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Table of Contents
Precalculus
Limits
2.5 Formal Definition of the Limit
2.6 Proofs of Some Basic Limit Rules
Differentiation
Basics of Differentiation
3.2 Product and Quotient Rules
3.3 Derivatives of Trigonometric Functions
3.5 Higher Order Derivatives: an introduction to second order derivatives
3.7 Derivatives of Exponential and Logarithm Functions
Applications of Derivatives
3.11 Extrema and Points of Inflection
Integration
Basics of Integration
4.2 Fundamental Theorem of Calculus
Integration Techniques
4.6 Derivative Rules and the Substitution Rule
4.8 Trigonometric Substitutions
4.10 Rational Functions by Partial Fraction Decomposition
4.11 Tangent Half Angle Substitution
Applications of Integration
4.18 Volume of solids of revolution
Parametric and Polar Equations
Parametric Equations
 Introduction to Parametric Equations
 Differentiation and Parametric Equations
 Integration and Parametric Equations
 Exercises
Polar Equations
Sequences and Series
Basics
Series and calculus
Exercises
Multivariable and Differential Calculus
Extensions
Advanced Integration Techniques
Further Analysis
Formal Theory of Calculus
Appendix
 Choosing delta
Solutions
 Precalculus/Solutions
 Infinity is not a number/Solutions
 Limits/Solutions
 Differentiation/Differentiation Defined/Solutions
 Chain Rule/Solutions
 Some Important Theorems/Solutions
 Differentiation/Basics of Differentiation/Solutions
 L'Hôpital's rule/Solutions
 Related Rates/Solutions
 Differentiation/Applications of Derivatives/Solutions
 Integration/Solutions
References
Acknowledgements and Further Reading
Introduction
What is calculus?
Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. As a typical application of the methods of calculus, consider a moving car. It is possible to create a function describing the displacement of the car (where it is located in relation to a reference point) at any point in time as well as a function describing the velocity (speed and direction of movement) of the car at any point in time. If the car were traveling at a constant velocity, then algebra would be sufficient to determine the position of the car at any time; if the velocity is unknown but still constant, the position of the car could be used (along with the time) to find the velocity.
However, the velocity of a car cannot jump from zero to 35 miles per hour at the beginning of a trip, stay constant throughout, and then jump back to zero at the end. As the accelerator is pressed down, the velocity rises gradually, and usually not at a constant rate (i.e., the driver may push on the gas pedal harder at the beginning, in order to speed up). Describing such motion and finding velocities and distances at particular times cannot be done using methods taught in precalculus, whereas it is not only possible but straightforward with calculus.
Calculus has two basic applications: differential calculus and integral calculus. The simplest introduction to differential calculus involves an explicit series of numbers. Given the series (42, 43, 3, 18, 34), the differential of this series would be (1, 40, 15, 16). The new series is derived from the difference of successive numbers which gives rise to its name "differential". Rarely, if ever, are differentials used on an explicit series of numbers as done here. Instead, they are derived from a continuous function in a manner which is described later.
Integral calculus, like differential calculus, can also be introduced via series of numbers. Notice that in the previous example, the original series can almost be derived solely from its differential. Instead of taking the difference, however, integration involves taking the sum. Given the first number of the original series, 42 in this case, the rest of the original series can be derived by adding each successive number in its differential (42+1, 4340, 3+15, 18+16). Note that knowledge of the first number in the original series is crucial in deriving the integral. As with differentials, integration is performed on continuous functions rather than explicit series of numbers, but the concept is still the same. Integral calculus allows us to calculate the area under a curve of almost any shape; in the car example, this enables you to find the displacement of the car based on the velocity curve. This is because the area under the curve is the total distance moved, as we will soon see. Let's understand this section very carefully. Suppose we have to add the numbers in series which is continuously "on" like 23,25,24,25,34,45,46,47, and so on...at this type integral calculation is very useful instead of the typical mathematical formulas.
Why learn calculus?
Calculus is essential for many areas of science and engineering. Both make heavy use of mathematical functions to describe and predict physical phenomena that are subject to continual change, and this requires the use of calculus. Take our car example: if you want to design cars, you need to know how to calculate forces, velocities, accelerations, and positions. All require calculus. Calculus is also necessary to study the motion of gases and particles, the interaction of forces, and the transfer of energy. It is also useful in business whenever rates are involved. For example, equations involving interest or supply and demand curves are grounded in the language of calculus.
Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and noneuclidean geometry.
Notwithstanding calculus' functional utility (pun intended), many nonscientists and nonengineers have chosen to study calculus just for the challenge of doing so. A smaller number of persons undertake such a challenge and then discover that calculus is beautiful in and of itself.
What is involved in learning calculus?
Learning calculus, like much of mathematics, involves two parts:
 Understanding the concepts: You must be able to explain what it means when you take a derivative rather than merely apply the formulas for finding a derivative. Otherwise, you will have no idea whether or not your solution is correct. Drawing diagrams, for example, can help clarify abstract concepts.
 Symbolic manipulation: Like other branches of mathematics, calculus is written in symbols that represent concepts. You will learn what these symbols mean and how to use them. A good working knowledge of trigonometry and algebra is a must, especially in integral calculus. Sometimes you will need to manipulate expressions into a usable form before it is possible to perform operations in calculus.
What you should know before using this text
There are some basic skills that you need before you can use this text. Continuing with our example of a moving car:
 You will need to describe the motion of the car in symbols. This involves understanding functions.
 You need to manipulate these functions. This involves algebra.
 You need to translate symbols into graphs and viceversa. This involves understanding the graphing of functions.
 It also helps (although it isn't necessarily essential) if you understand the functions used in trigonometry since these functions appear frequently in science.
Scope
The first four chapters of this textbook cover the topics taught in a typical high school or first year college course. The first chapter, Precalculus, reviews those aspects of functions most essential to the mastery of calculus. The second, Limits, introduces the concept of the limit process. It also discusses some applications of limits and proposes using limits to examine slope and area of functions. The next two chapters, Differentiation and Integration, apply limits to calculate derivatives and integrals. The Fundamental Theorem of Calculus is used, as are the essential formulae for computation of derivatives and integrals without resorting to the limit process. The third and fourth chapters include articles that apply the concepts previously learned to calculating volumes, and so on as well as other important formulae.
The remainder of the central Calculus chapters cover topics taught in higherlevel calculus topics: multivariable calculus, vectors, and series (Taylor, convergent, divergent).
Finally, the other chapters cover the same material, using formal notation. They introduce the material at a much faster pace, and cover many more theorems than the other two sections. They assume knowledge of some set theory and set notation.
Precalculus
<h1> 1.1 Algebra</h1>
This section is intended to review algebraic manipulation. It is important to understand algebra in order to do calculus. If you have a good knowledge of algebra, you should probably just skim this section to be sure you are familiar with the ideas.
Rules of arithmetic and algebra
The following laws are true for all a, b, and c, whether a, b, and c are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.
Addition
 Commutative Law: .
 Associative Law: .
 Additive Identity: .
 Additive Inverse: .
Subtraction
 Definition: .
Multiplication
 Commutative law: .
 Associative law: .
 Multiplicative identity: .
 Multiplicative inverse: , whenever
 Distributive law: .
Division
 Definition: , whenever .
Let's look at an example to see how these rules are used in practice.
= (from the definition of division)  
= (from the associative law of multiplication)  
= (from multiplicative inverse)  
= (from multiplicative identity) 
Of course, the above is much longer than simply cancelling out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:
 .
The correct simplification is
 ,
where the number cancels out in both the numerator and the denominator.
Interval notation
There are a few different ways that one can express with symbols a specific interval (all the numbers between two numbers). One way is with inequalities. If we wanted to denote the set of all numbers between, say, 2 and 4, we could write "all x satisfying 2<x<4." This excludes the endpoints 2 and 4 because we use < instead of . If we wanted to include the endpoints, we would write "all x satisfying ." This includes the endpoints.
Another way to write these intervals would be with interval notation. If we wished to convey "all x satisfying 2<x<4" we would write (2,4). This does not include the endpoints 2 and 4. If we wanted to include the endpoints we would write [2,4]. If we wanted to include 2 and not 4 we would write [2,4); if we wanted to exclude 2 and include 4, we would write (2,4].
Thus, we have the following table:
Endpoint conditions  Inequality notation  Interval notation 

Including both 2 and 4  all x satisfying 

Not including 2 nor 4  all x satisfying 

Including 2 not 4  all x satisfying 

Including 4 not 2  all x satisfying 

In general, we have the following table:
Meaning  Interval Notation  Set Notation 

All values greater than or equal to and less than or equal to  
All values greater than and less than  
All values greater than or equal to and less than  
All values greater than and less than or equal to  
All values greater than or equal to .  
All values greater than .  
All values less than or equal to .  
All values less than .  
All values. 
Note that and must always have an exclusive parenthesis rather than an inclusive bracket. This is because is not a number, and therefore cannot be in our set. is really just a symbol that makes things easier to write, like the intervals above.
The interval (a,b) is called an open interval, and the interval [a,b] is called a closed interval.
Intervals are sets and we can use set notation to show relations between values and intervals. If we want to say that a certain value is contained in an interval, we can use the symbol to denote this. For example, . Likewise, the symbol denotes that a certain element is not in an interval. For example .
Exponents and radicals
There are a few rules and properties involving exponents and radicals that you'd do well to remember. As a definition we have that if n is a positive integer then denotes n factors of a. That is,
If then we say that .
If n is a negative integer then we say that
If we have an exponent that is a fraction then we say that
In addition to the previous definitions, the following rules apply:
Rule  Example 

Factoring and roots
Given the expression , one may ask "what are the values of x that make this expression 0?" If we factor we obtain
If x=1 or 2, then one of the factors on the right becomes zero. Therefore, the whole must be zero. So, by factoring we have discovered the values of x that render the expression zero. These values are termed "roots." In general, given a quadratic polynomial that factors as
then we have that x = c/a and x = d/b are roots of the original polynomial.
A special case to be on the look out for is the difference of two squares, . In this case, we are always able to factor as
For example, consider . On initial inspection we would see that both and are squares ( and ). Applying the previous rule we have
The following is a general result of great utility.
The quadratic formula
Given any quadratic equation , all solutions of the equation are given by the quadratic formula:
Example: Find all the roots of
Finding the roots is equivalent to solving the equation . Applying the quadratic formula with , we have: 
The quadratic formula can also help with factoring, as the next example demonstrates.
Example: Factor the polynomial
We already know from the previous example that the polynomial has roots and . Our factorization will take the form 
Note that if then the roots will not be real numbers.
Simplifying rational expressions
Consider the two polynomials
and
When we take the quotient of the two we obtain
The ratio of two polynomials is called a rational expression. Many times we would like to simplify such a beast. For example, say we are given We may simplify this in the following way:
This is nice because we have obtained something we understand quite well, , from something we didn't.
Formulas of multiplication of polynomials
Here are some formulas that can be quite useful for solving polynomial problems:
Polynomial Long Division
Suppose we would like to divide one polynomial by another. The procedure is similar to long division of numbers and is illustrated in the following example:
Example
Divide (the dividend or numerator) by (the divisor or denominator)
Similar to long division of numbers, we set up our problem as follows: First we have to answer the question, how many times does go into ? To find out, divide the leading term of the dividend by leading term of the divisor. So it goes in times. We record this above the leading term of the dividend: , and we multiply by and write this below the dividend as follows: Now we perform the subtraction, bringing down any terms in the dividend that aren't matched in our subtrahend: Now we repeat, treating the bottom line as our new dividend: In this case we have no remainder. 
Application: Factoring Polynomials
We can use polynomial long division to factor a polynomial if we know one of the factors in advance. For example, suppose we have a polynomial and we know that is a root of . If we perform polynomial long division using P(x) as the dividend and as the divisor, we will obtain a polynomial such that , where the degree of is one less than the degree of .
Exercise
Application: Breaking up a rational function
Similar to the way one can convert an improper fraction into an integer plus a proper fraction, one can convert a rational function whose numerator has degree and whose denominator has degree with into a polynomial plus a rational function whose numerator has degree and denominator has degree with .
Suppose that divided by has quotient and remainder . That is
Dividing both sides by gives
will have degree less than .
Example
Write as a polynomial plus a rational function with numerator having degree less than the denominator.
so 
<h1> 1.2 Functions</h1>
What functions are and how are they described
Note: This is an attempt at a rewrite of "Classical understanding of functions". If others approve, consider deleting that section.
Whenever one quantity uniquely determines the value of another quantity, we have a function.
{  comments 
ie X uniquely determines Y but Y is not uniquely determined by X
Let set X consists of x's and set Y consists of y's
two x's can have same y ie one y can be determined by two x's
but one x cannot have two y's
 end of comments 
}
You can think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product.
A function in everyday life
Think about dropping a ball from a bridge. At each moment in time, the ball is a height above the ground. The height of the ball is a function of time. It was the job of physicists to come up with a formula for this function. This type of function is called realvalued since the "finished product" is a number (or, more specifically, a real number). 
A function in everyday life (Preview of Multivariable Calculus)
Think about a wind storm. At different places, the wind can be blowing in different directions with different intensities. The direction and intensity of the wind can be thought of as a function of position. This is a function of two real variables (a location is described by two values  an and a ) which results in a vector (which is something that can be used to hold a direction and an intensity). These functions are studied in multivariable calculus (which is usually studied after a one year college level calculus course).This a vectorvalued function of two real variables. 
We will be looking at realvalued functions until studying multivariable calculus. Think of a realvalued function as an inputoutput machine; you give the function an input, and it gives you an output which is a number (more specifically, a real number). For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input 1 and gives the output value 1.
There are many ways which people describe functions. In the examples above, a verbal descriptions is given (the height of the ball above the earth as a function of time). Here is a list of ways to describe functions. The top three listed approaches to describing functions are the most popular and you could skip the rest if you like.
 A function is given a name (such as ) and a formula for the function is also given. For example, describes a function. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument.
 A function is described using an equation and two variables. One variable is for the input of the function and one is for the output of the function. The variable for the input is called the independent variable. The variable for the output is called the dependent variable. For example, describes a function. The dependent variable appears by itself on the left hand side of equal sign.
 A verbal description of the function.
When a function is given a name (like in number 1 above), the name of the function is usually a single letter of the alphabet (such as or ). Some functions whose names are multiple letters (like the sine function .
Plugging a value into a function
If we write , then we know that
How would we know the value of the function at 3? We would have the following three thoughts: and we would write . The value of at 3 is 11. Note that means the value of the dependent variable when takes on the value of 3. So we see that the number 11 is the output of the function when we give the number 3 as the input. People often summarize the work above by writing "the value of at three is eleven", or simply " of three equals eleven". 
Classical understanding of functions
To provide the classical understanding of functions, think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product based on a specific set of instructions. The kinds of functions we consider here, for the most part, take in a real number, change it in a formulaic way, and give out a real number (possibly the same as the one it took in). Think of this as an inputoutput machine; you give the function an input, and it gives you an output. For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input and gives the output value 1.
A function is usually written as , , or something similar  although it doesn't have to be. A function is always defined as "of a variable" which tells us what to replace in the formula for the function.
For example, tells us:
 The function is a function of .
 To evaluate the function at a certain number, replace the with that number.
 Replacing with that number in the right side of the function will produce the function's output for that certain input.
 In English, the definition of is interpreted, "Given a number, will return two more than the triple of that number."
Thus, if we want to know the value (or output) of the function at 3:
 We evaluate the function at .
 The value of at 3 is 11.
See? It's easy!
Note that means the value of the dependent variable when takes on the value of 3. So we see that the number 11 is the output of the function when we give the number 3 as the input. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument (or the dependent variable). A good way to think of it is the dependent variable 'depends' on the value of the independent variable . This is read as "the value of at three is eleven", or simply " of three equals eleven".
Notation
Functions are used so much that there is a special notation for them. The notation is somewhat ambiguous, so familiarity with it is important in order to understand the intention of an equation or formula.
Though there are no strict rules for naming a function, it is standard practice to use the letters , , and to denote functions, and the variable to denote an independent variable. is used for both dependent and independent variables.
When discussing or working with a function , it's important to know not only the function, but also its independent variable . Thus, when referring to a function , you usually do not write , but instead . The function is now referred to as " of ". The name of the function is adjacent to the independent variable (in parentheses). This is useful for indicating the value of the function at a particular value of the independent variable. For instance, if
 ,
and if we want to use the value of for equal to , then we would substitute 2 for on both sides of the definition above and write
This notation is more informative than leaving off the independent variable and writing simply '', but can be ambiguous since the parentheses can be misinterpreted as multiplication.
Modern understanding of functions
The formal definition of a function states that a function is actually a rule that associates elements of one set called the domain of the function, with the elements of another set called the range of the function. For each value we select from the domain of the function, there exists exactly one corresponding element in the range of the function. The definition of the function tells us which element in the range corresponds to the element we picked from the domain. Classically, the element picked from the domain is pictured as something that is fed into the function and the corresponding element in the range is pictured as the output. Since we "pick" the element in the domain whose corresponding element in the range we want to find, we have control over what element we pick and hence this element is also known as the "independent variable". The element mapped in the range is beyond our control and is "mapped to" by the function. This element is hence also known as the "dependent variable", for it depends on which independent variable we pick. Since the elementary idea of functions is better understood from the classical viewpoint, we shall use it hereafter. However, it is still important to remember the correct definition of functions at all times.
To make it simple, for the function , all of the possible values constitute the domain, and all of the values ( on the xy plane) constitute the range.
Remarks
The following arise as a direct consequence of the definition of functions:
 By definition, for each "input" a function returns only one "output", corresponding to that input. While the same output may correspond to more than one input, one input cannot correspond to more than one output. This is expressed graphically as the vertical line test: a line drawn parallel to the axis of the dependent variable (normally vertical) will intersect the graph of a function only once. However, a line drawn parallel to the axis of the independent variable (normally horizontal) may intersect the graph of a function as many times as it likes. Equivalently, this has an algebraic (or formulabased) interpretation. We can always say if , then , but if we only know that then we can't be sure that .
 Each function has a set of values, the function's domain, which it can accept as input. Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}. This set must be implicitly/explicitly defined in the definition of the function. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element.
 Each function has a set of values, the function's range, which it can output. This may be the set of real numbers. It may be the set of positive integers or even the set {0,1}. This set, too, must be implicitly/explicitly defined in the definition of the function.
The vertical line test
The vertical line test, mentioned in the preceding paragraph, is a systematic test to find out if an equation involving and can serve as a function (with the independent variable and the dependent variable). Simply graph the equation and draw a vertical line through each point of the axis. If any vertical line ever touches the graph at more than one point, then the equation is not a function; if the line always touches at most one point of the graph, then the equation is a function.
(There are a lot of useful curves, like circles, that aren't functions (see picture). Some people call these graphs with multiple intercepts, like our circle, "multivalued functions"; they would refer to our "functions" as "singlevalued functions".)
Important functions
Constant function 
It disregards the input and always outputs the constant , and is a polynomial of the zeroth degree where f(x) = cx^{0}= c(1) = c. Its graph is a horizontal line. 

Linear function 
Takes an input, multiplies by m and adds c. It is a polynomial of the first degree. Its graph is a line (slanted, except ). 

Identity function 
Takes an input and outputs it unchanged. A polynomial of the first degree, f(x) = x^{1} = x. Special case of a linear function. 

Quadratic function 
A polynomial of the second degree. Its graph is a parabola, unless . (Don't worry if you don't know what this is.) 

Polynomial function 
The number is called the degree. 

Signum function 
Determines the sign of the argument . 
Example functions
Some more simple examples of functions have been listed below.



It is possible to replace the independent variable with any mathematical expression, not just a number. For instance, if the independent variable is itself a function of another variable, then it could be replaced with that function. This is called composition, and is discussed later.
Manipulating functions
Addition, Subtraction, Multiplication and Division of functions
For two realvalued functions, we can add the functions, multiply the functions, raised to a power, etc.
Example: Adding, subtracting, multiplying and dividing functions which do not have a name
If we add the functions and , we obtain .

If a math problem wants you to add two functions and , there are two ways that the problem will likely be worded:
 If you are told that , that , that and asked about , then you are being asked to add two functions. Your answer would be .
 If you are told that , that and you are asked about , then you are being asked to add two functions. The addition of and is called . Your answer would be .
Similar statements can be made for subtraction, multiplication and division.
Example: Adding, subtracting, multiplying and dividing functions which do have a name
Let and:. Let's add, subtract, multiply and divide.

Composition of functions
We begin with a fun (and not too complicated) application of composition of functions before we talk about what composition of functions is.
Example: Dropping a ball
If we drop a ball from a bridge which is 20 meters above the ground, then the height of our ball above the earth is a function of time. The physicists tell us that if we measure time in seconds and distance in meters, then the formula for height in terms of time is . Suppose we are tracking the ball with a camera and always want the ball to be in the center of our picture. Suppose we have The angle will depend upon the height of the ball above the ground and the height above the ground depends upon time. So the angle will depend upon time. This can be written as . We replace with what it is equal to. This is the essence of composition. 
Composition of functions is another way to combine functions which is different from addition, subtraction, multiplication or division.
The value of a function depends upon the value of another variable ; however, that variable could be equal to another function , so its value depends on the value of a third variable. If this is the case, then the first variable is a function of the third variable; this function () is called the composition of the other two functions ( and ).
Example: Composing two functions
Let and:. The composition of with is read as either "f composed with g" or "f of g of x." Let Then
Sometimes a math problem asks you compute when they want you to compute , Here, is the composition of and and we write . Note that composition is not commutative:

Composition of functions is very common, mainly because functions themselves are common. For instance, squaring and sine are both functions:
 ,
Thus, the expression is a composition of functions:

= = .
(Note that this is not the same as .) Since the function sine equals if ,
 .
Since the function square equals if ,
 .
Transformations
Transformations are a type of function manipulation that are very common. They consist of multiplying, dividing, adding or subtracting constants to either the input or the output. Multiplying by a constant is called dilation and adding a constant is called translation. Here are a few examples:
 Dilation
 Translation
 Dilation
 Translation
Translations and dilations can be either horizontal or vertical. Examples of both vertical and horizontal translations can be seen at right. The red graphs represent functions in their 'original' state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphs have been translated vertically.
Dilations are demonstrated in a similar fashion. The function
has had its input doubled. One way to think about this is that now any change in the input will be doubled. If I add one to , I add two to the input of , so it will now change twice as quickly. Thus, this is a horizontal dilation by because the distance to the axis has been halved. A vertical dilation, such as
is slightly more straightforward. In this case, you double the output of the function. The output represents the distance from the axis, so in effect, you have made the graph of the function 'taller'. Here are a few basic examples where is any positive constant:
Original graph  Rotation about origin  
Horizontal translation by units left  Horizontal translation by units right  
Horizontal dilation by a factor of  Vertical dilation by a factor of  
Vertical translation by units down  Vertical translation by units up  
Reflection about axis  Reflection about axis 
Domain and Range
Domain
The domain of a function is the set of all points over which it is defined. More simply, it represents the set of xvalues which the function can accept as input. For instance, if
then is only defined for values of between and , because the square root function is not defined (in real numbers) for negative values. Thus, the domain, in interval notation, is . In other words,
 .
Range
The range of a function is the set of all values which it attains (i.e. the yvalues). For instance, if:
 ,
then can only equal values in the interval from to . Thus, the range of is .
Onetoone Functions
A function is onetoone (or less commonly injective) if, for every value of , there is only one value of that corresponds to that value of . For instance, the function is not onetoone, because both and result in . However, the function is onetoone, because, for every possible value of , there is exactly one corresponding value of . Other examples of onetoone functions are , where . Note that if you have a onetoone function and translate or dilate it, it remains onetoone. (Of course you can't multiply or by a zero factor).
Horizontal Line Test
If you know what the graph of a function looks like, it is easy to determine whether or not the function is onetoone. If every horizontal line intersects the graph in at most one point, then the function is onetoone. This is known as the Horizontal Line Test.
Algebraic 11 Test
You can also show onetooneness algebraically by assuming that two inputs give the same output and then showing that the two inputs must have been equal. For example, Is a 11 function?
Therefore by the algebraic 11 test, the function is 11.
You can show that a function is not onetoone by finding two distinct inputs that give the same output. For example, is not onetoone because but .
Inverse functions
We call the inverse function of if, for all :
 .
A function has an inverse function if and only if is onetoone. For example, the inverse of is . The function has no inverse.
Notation
The inverse function of is denoted as . Thus, is defined as the function that follows this rule
:
To determine when given a function , substitute for and substitute for . Then solve for , provided that it is also a function.
Example: Given , find .
Substitute for and substitute for . Then solve for :
To check your work, confirm that :
If isn't onetoone, then, as we said before, it doesn't have an inverse. Then this method will fail.
Example: Given , find .
Substitute for and substitute for . Then solve for :
Since there are two possibilities for , it's not a function. Thus doesn't have an inverse. Of course, we could also have found this out from the graph by applying the Horizontal Line Test. It's useful, though, to have lots of ways to solve a problem, since in a specific case some of them might be very difficult while others might be easy. For example, we might only know an algebraic expression for but not a graph.
External links
<h1> 1.3 Graphing linear functions</h1>
It is sometimes difficult to understand the behavior of a function given only its definition; a visual representation or graph can be very helpful. A graph is a set of points in the Cartesian plane, where each point (,) indicates that . In other words, a graph uses the position of a point in one direction (the verticalaxis or yaxis) to indicate the value of for a position of the point in the other direction (the horizontalaxis or xaxis).
Functions may be graphed by finding the value of for various and plotting the points (, ) in a Cartesian plane. For the functions that you will deal with, the parts of the function between the points can generally be approximated by drawing a line or curve between the points. Extending the function beyond the set of points is also possible, but becomes increasingly inaccurate.
Example
Plotting points like this is laborious. Fortunately, many functions' graphs fall into general patterns. For a simple case, consider functions of the form
The graph of is a single line, passing through the point with slope 3. Thus, after plotting the point, a straightedge may be used to draw the graph. This type of function is called linear and there are a few different ways to present a function of this type.
Slopeintercept form
When we see a function presented as
we call this presentation the slopeintercept form. This is because, not surprisingly, this way of writing a linear function involves the slope, m, and the yintercept, b.
Pointslope form
If someone walks up to you and gives you one point and a slope, you can draw one line and only one line that goes through that point and has that slope. Said differently, a point and a slope uniquely determine a line. So, if given a point and a slope m, we present the graph as
We call this presentation the pointslope form. The pointslope and slopeintercept form are essentially the same. In the pointslope form we can use any point the graph passes through. Where as, in the slopeintercept form, we use the yintercept, that is the point (0,b).
Calculating slope
If given two points, and , we may then compute the slope of the line that passes through these two points. Remember, the slope is determined as "rise over run." That is, the slope is the change in yvalues divided by the change in xvalues. In symbols,
So now the question is, "what's and ?" We have that and . Thus,
Twopoint form
Two points also uniquely determine a line. Given points and , we have the equation
This presentation is in the twopoint form. It is essentially the same as the pointslope form except we substitute the expression for m.
<h1> 1.4 Precalculus Cumulative Exercises</h1>
Algebra
Convert to interval notation
State the following intervals using set notation
Which one of the following is a true statement?
Hint: the true statement is often referred to as the triangle inequality. Give examples where the other two are false.
Evaluate the following expressions
Simplify the following
Find the roots of the following polynomials
Factor the following expressions
Simplify the following
Functions
52. Let .
53. Let , .
 a. Give formulae for
55. Consider the following function
56. Consider the following function
57. Consider the following function
58. Consider the following function
Graphing
Limits
<h1> 2.1 An Introduction to Limits</h1>
Intuitive Look
A limit looks at what happens to a function when the input approaches a certain value. The general notation for a limit is as follows:
This is read as "The limit of of as approaches ". We'll take up later the question of how we can determine whether a limit exists for at and, if so, what it is. For now, we'll look at it from an intuitive standpoint.
Let's say that the function that we're interested in is , and that we're interested in its limit as approaches . Using the above notation, we can write the limit that we're interested in as follows:
One way to try to evaluate what this limit is would be to choose values near 2, compute for each, and see what happens as they get closer to 2. This is implemented as follows:
1.7  1.8  1.9  1.95  1.99  1.999  
2.89  3.24  3.61  3.8025  3.9601  3.996001 
Here we chose numbers smaller than 2, and approached 2 from below. We can also choose numbers larger than 2, and approach 2 from above:
2.3  2.2  2.1  2.05  2.01  2.001  
5.29  4.84  4.41  4.2025  4.0401  4.004001 
We can see from the tables that as grows closer and closer to 2, seems to get closer and closer to 4, regardless of whether approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of as approaches 2 is 4, or, written in limit notation,
We could have also just substituted 2 into and evaluated: . However, this will not work with all limits.
Now let's look at another example. Suppose we're interested in the behavior of the function as approaches 2. Here's the limit in limit notation:
Just as before, we can compute function values as approaches 2 from below and from above. Here's a table, approaching from below:
1.7  1.8  1.9  1.95  1.99  1.999  
3.333  5  10  20  100  1000 
And here from above:
2.3  2.2  2.1  2.05  2.01  2.001  
3.333  5  10  20  100  1000 
In this case, the function doesn't seem to be approaching a single value as approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). This is known as an infinite limit. Note that we cannot just substitute 2 into and evaluate as we could with the first example, since we would be dividing by 0.
Both of these examples may seem trivial, but consider the following function:
This function is the same as
Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input.
In elementary algebra, a typical approach is to simply say that we can cancel the term , and then we have the function . However, that would be inaccurate; the function that we have now is not really the same as the one we started with, because it is defined when , and our original function was specifically not defined when . This may seem like a minor point, but from making this kind of assumptions we can easily derive absurd results, such that (see Mathematical Fallacy § All numbers equal all other numbers in Wikipedia for a complete example). Even without calculus we can avoid this error by stating that:
In calculus, we can introduce a more intuitive and also correct way of looking at this type of function. What we want is to be able to say that, although the function isn't defined when , it works almost as if it was. It may not get there, but it gets really, really close. For instance, . The only question that we have is: what do we mean by "close"?
Informal Definition of a Limit
As the precise definition of a limit is a bit technical, it is easier to start with an informal definition; we'll explain the formal definition later.
We suppose that a function is defined for near (but we do not require that it be defined when ).
We call the limit of as approaches if becomes close to when is close (but not equal) to , and if there is no other value with the same property..
When this holds we write
or
Notice that the definition of a limit is not concerned with the value of when (which may exist or may not). All we care about are the values of when is close to , on either the left or the right (i.e. less or greater).
Limit Rules
Now that we have defined, informally, what a limit is, we will list some rules that are useful for working with and computing limits. You will be able to prove all these once we formally define the fundamental concept of the limit of a function.
First, the constant rule states that if (that is, is constant for all ) then the limit as approaches must be equal to . In other words
Constant Rule for Limits
 If b and c are constants then .
 Example:
Second, the identity rule states that if (that is, just gives back whatever number you put in) then the limit of as approaches is equal to . That is,
Identity Rule for Limits
 If c is a constant then .
 Example:
The next few rules tell us how, given the values of some limits, to compute others.
Operational Identities for Limits
Suppose that and and that is constant. Then
Notice that in the last rule we need to require that is not equal to zero (otherwise we would be dividing by zero which is an undefined operation).
These rules are known as identities; they are the scalar product, sum, difference, product, and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by a constant, we say that you are performing scalar multiplication.)
Using these rules we can deduce another. Namely, using the rule for products many times we get that
 for a positive integer .
This is called the power rule.
Examples
 Example 1
Find the limit .
We need to simplify the problem, since we have no rules about this expression by itself. We know from the identity rule above that . By the power rule, . Lastly, by the scalar multiplication rule, we get .
 Example 2
Find the limit .
To do this informally, we split up the expression, once again, into its components. As above,.
Also and . Adding these together gives
 .
 Example 3
Find the limit .
From the previous example the limit of the numerator is . The limit of the denominator is
As the limit of the denominator is not equal to zero we can divide. This gives
 .
 Example 4
Find the limit .
We apply the same process here as we did in the previous set of examples;
 .
We can evaluate each of these; and Thus, the answer is .
 Example 5
Find the limit .
In this example, evaluating the result directly will result in a division by zero. While you can determine the answer experimentally, a mathematical solution is possible as well.
First, the numerator is a polynomial that may be factored:
Now, you can divide both the numerator and denominator by (x2):
 Example 6
Find the limit .
To evaluate this seemingly complex limit, we will need to recall some sine and cosine identities. We will also have to use two new facts. First, if is a trigonometric function (that is, one of sine, cosine, tangent, cotangent, secant or cosecant) and is defined at , then .
Second, . This may be determined experimentally, or by applying L'Hôpital's rule, described later in the book.
To evaluate the limit, recognize that can be multiplied by to obtain which, by our trig identities, is . So, multiply the top and bottom by . (This is allowed because it is identical to multiplying by one.) This is a standard trick for evaluating limits of fractions; multiply the numerator and the denominator by a carefully chosen expression which will make the expression simplify somehow. In this case, we should end up with:
.
Our next step should be to break this up into by the product rule. As mentioned above, .
Next, .
Thus, by multiplying these two results, we obtain 0.
We will now present an amazingly useful result, even though we cannot prove it yet. We can find the limit at of any polynomial or rational function, as long as that rational function is defined at (so we are not dividing by zero). That is, must be in the domain of the function.
If is a polynomial or rational function that is defined at then
We already learned this for trigonometric functions, so we see that it is easy to find limits of polynomial, rational or trigonometric functions wherever they are defined. In fact, this is true even for combinations of these functions; thus, for example, .
The Squeeze Theorem
The Squeeze Theorem is very important in calculus, where it is typically used to find the limit of a function by comparison with two other functions whose limits are known.
It is called the Squeeze Theorem because it refers to a function whose values are squeezed between the values of two other functions and , both of which have the same limit . If the value of is trapped between the values of the two functions and , the values of must also approach .
Expressed more precisely:
Suppose that holds for all in some open interval containing , except possibly at itself. Suppose also that . Then also.
Example: Compute . Note that the sine of any real number is in the interval . That is, for all , and for all . If is positive, we can multiply these inequalities by and get . If is negative, we can similarly multiply the inequalities by the positive number and get . Putting these together, we can see that, for all nonzero , . But it's easy to see that . So, by the Squeeze Theorem, .
Finding Limits
Now, we will discuss how, in practice, to find limits. First, if the function can be built out of rational, trigonometric, logarithmic and exponential functions, then if a number is in the domain of the function, then the limit at is simply the value of the function at .
If is not in the domain of the function, then in many cases (as with rational functions) the domain of the function includes all the points near , but not itself. An example would be if we wanted to find , where the domain includes all numbers besides 0.
In that case, in order to find we want to find a function similar to , except with the hole at filled in. The limits of and will be the same, as can be seen from the definition of a limit. By definition, the limit depends on only at the points where is close to but not equal to it, so the limit at does not depend on the value of the function at . Therefore, if , also. And since the domain of our new function includes , we can now (assuming is still built out of rational, trigonometric, logarithmic and exponential functions) just evaluate it at as before. Thus we have .
In our example, this is easy; canceling the 's gives , which equals at all points except 0. Thus, we have . In general, when computing limits of rational functions, it's a good idea to look for common factors in the numerator and denominator.
Lastly, note that the limit might not exist at all. There are a number of ways in which this can occur:
 "Gap"
 There is a gap (not just a single point) where the function is not defined. As an example, in
 does not exist when . There is no way to "approach" the middle of the graph. Note that the function also has no limit at the endpoints of the two curves generated (at and ). For the limit to exist, the point must be approachable from both the left and the right.
 Note also that there is no limit at a totally isolated point on a graph.
 "Jump"
 If the graph suddenly jumps to a different level, there is no limit at the point of the jump. For example, let be the greatest integer . Then, if is an integer, when approaches from the right , while when approaches from the left . Thus will not exist.
 Vertical asymptote
 In
 the graph gets arbitrarily high as it approaches 0, so there is no limit. (In this case we sometimes say the limit is infinite; see the next section.)
 Infinite oscillation
 These next two can be tricky to visualize. In this one, we mean that a graph continually rises above and falls below a horizontal line. In fact, it does this infinitely often as you approach a certain value. This often means that there is no limit, as the graph never approaches a particular value. However, if the height (and depth) of each oscillation diminishes as the graph approaches the value, so that the oscillations get arbitrarily smaller, then there might actually be a limit.
 The use of oscillation naturally calls to mind the trigonometric functions. An example of a trigonometric function that does not have a limit as approaches 0 is
 As gets closer to 0 the function keeps oscillating between and 1. In fact, oscillates an infinite number of times on the interval between 0 and any positive value of . The sine function is equal to zero whenever , where is a positive integer. Between every two integers , goes back and forth between 0 and or 0 and 1. Hence, for every . In between consecutive pairs of these values, and , goes back and forth from 0, to either or 1 and back to 0. We may also observe that there are an infinite number of such pairs, and they are all between 0 and . There are a finite number of such pairs between any positive value of and , so there must be infinitely many between any positive value of and 0. From our reasoning we may conclude that, as approaches 0 from the right, the function does not approach any specific value. Thus, does not exist.
Using Limit Notation to Describe Asymptotes
Now consider the function
What is the limit as approaches zero? The value of does not exist; it is not defined.
Notice, also, that we can make as large as we like, by choosing a small , as long as . For example, to make equal to , we choose to be . Thus, does not exist.
However, we do know something about what happens to when gets close to 0 without reaching it. We want to say we can make arbitrarily large (as large as we like) by taking to be sufficiently close to zero, but not equal to zero. We express this symbolically as follows:
Note that the limit does not exist at ; for a limit, being is a special kind of not existing. In general, we make the following definition.
We say the limit of as approaches is infinity if becomes very big (as big as we like) when is close (but not equal) to .
In this case we write
or
 .
Similarly, we say the limit of as approaches is negative infinity if becomes very negative when is close (but not equal) to .
In this case we write
or
 .
An example of the second half of the definition would be that .
Key Application of Limits
To see the power of the concept of the limit, let's consider a moving car. Suppose we have a car whose position is linear with respect to time (that is, a graph plotting the position with respect to time will show a straight line). We want to find the velocity. This is easy to do from algebra; we just take the slope, and that's our velocity.
But unfortunately, things in the real world don't always travel in nice straight lines. Cars speed up, slow down, and generally behave in ways that make it difficult to calculate their velocities.
Now what we really want to do is to find the velocity at a given moment (the instantaneous velocity). The trouble is that in order to find the velocity we need two points, while at any given time, we only have one point. We can, of course, always find the average speed of the car, given two points in time, but we want to find the speed of the car at one precise moment.
This is the basic trick of differential calculus, the first of the two main subjects of this book. We take the average speed at two moments in time, and then make those two moments in time closer and closer together. We then see what the limit of the slope is as these two moments in time are closer and closer, and say that this limit is the slope at a single instant.
We will study this process in much greater depth later in the book. First, however, we will need to study limits more carefully.
External Links
<h1> 2.2 Finite Limits</h1>
Informal Finite Limits
Now, we will try to more carefully restate the ideas of the last chapter. We said then that the equation meant that, when gets close to 2, gets close to 4. What exactly does this mean? How close is "close"? The first way we can approach the problem is to say that, at , , which is pretty close to 4.
Sometimes however, the function might do something completely different. For instance, suppose , so . Next, if you take a value even closer to 2, , in this case you actually move further from 4. The reason for this is that substitution gives us 4.23 as x approaches 2.
The solution is to find out what happens arbitrarily close to the point. In particular, we want to say that, no matter how close we want the function to get to 4, if we make close enough to 2 then it will get there. In this case, we will write
and say "The limit of , as approaches 2, equals 4" or "As approaches 2, approaches 4." In general:
We call the limit of as approaches if becomes arbitrarily close to whenever is sufficiently close (and not equal) to .
When this holds we write
or
OneSided Limits
Sometimes, it is necessary to consider what happens when we approach an value from one particular direction. To account for this, we have onesided limits. In a lefthanded limit, approaches from the lefthand side. Likewise, in a righthanded limit, approaches from the righthand side.
For example, if we consider , there is a problem because there is no way for to approach 2 from the left hand side (the function is undefined here). But, if approaches 2 only from the righthand side, we want to say that approaches 0.
We call the limit of as approaches from the right if becomes arbitrarily close to whenever is sufficiently close to and greater than .
When this holds we write
Similarly, we call the limit of as approaches from the left if becomes arbitrarily close to whenever is sufficiently close to and less than .
When this holds we write
In our example, the lefthanded limit does not exist.
The righthanded limit, however, .
It is a fact that exists if and only if and exist and are equal to each other. In this case, will be equal to the same number.
In our example, one limit does not even exist. Thus does not exist either.
<h1> 2.3 Infinite Limits</h1>
Informal Infinite Limits
Another kind of limit involves looking at what happens to as gets very big. For example, consider the function . As gets very big, gets very small. In fact, gets closer and closer to zero the bigger gets. Without limits it is very difficult to talk about this fact, because can keep getting bigger and bigger and never actually gets to zero; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something  without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does have to be to be sure that is really going towards 0?
In this case, we want to say that, however close we want to get to 0, for big enough is guaranteed to get that close. So we have yet another definition.
We call the limit of as approaches infinity if becomes arbitrarily close to whenever is sufficiently large.
When this holds we write
or
Similarly, we call the limit of as approaches negative infinity if becomes arbitrarily close to whenever is sufficiently negative.
When this holds we write
or
So, in this case, we write:
and say "The limit, as approaches infinity, equals ," or "as approaches infinity, the function approaches ".
We can also write:
because making very negative also forces to be close to .
Notice, however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.
Limits at Infinity of Rational Functions
One special case that comes up frequently is when we want to find the limit at (or ) of a rational function. A rational function is just one made by dividing two polynomials by each other. For example, is a rational function. Also, any polynomial is a rational function, since is just a (very simple) polynomial, so we can write the function as , the quotient of two polynomials.
Consider the numerator of a rational function as we allow the variable to grow very large (in either the positive or negative sense). The term with the highest exponent on the variable will dominate the numerator, and the other terms become more and more insignificant compared to the dominating term. The same applies to the denominator. In the limit, the other terms become negligible, and we only need to examine the dominating term in the numerator and denominator.
There is a simple rule for determining a limit of a rational function as the variable approaches infinity. Look for the term with the highest exponent on the variable in the numerator. Look for the same in the denominator. This rule is based on that information.
 If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit (at both and ) is the ratio of the coefficients of the highest terms.
 If the numerator has the highest term, then the fraction is called "topheavy". If, when you divide the numerator by the denominator the resulting exponent on the variable is even, then the limit (at both and ) is . If it is odd, then the limit at is , and the limit at is .
 If the denominator has the highest term, then the fraction is called "bottomheavy" and the limit (at both and ) is zero.
Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same as . Also, a straight power of , like , has coefficient 1, since it is the same as .
Examples
 Example 1
Find .
The function is the quotient of two polynomials, and . By our rule we look for the term with highest exponent in the numerator; it's . The term with highest exponent in the denominator is also . So, the limit is the ratio of their coefficients. Since , both coefficients are 1, .
 Example 2
Find .
We look at the terms with the highest exponents; for the numerator, it is , while for the denominator it is . Since the exponent on the numerator is higher, we know the limit at will be . So,
.
Infinity is not a number
Most people seem to struggle with this fact when first introduced to calculus, and in particular limits.
But is different. is not a number.
Mathematics is based on formal rules that govern the subject. When a list of formal rules applies to a type of object (e.g., "a number") those rules must always apply — no exceptions!
What makes different is this: "there is no number greater than infinity". You can write down the formula in a lot of different ways, but here's one way: . If you add one to infinity, you still have infinity; you don't have a bigger number. If you believe that, then infinity is not a number.
Since does not follow the rules laid down for numbers, it cannot be a number. Every time you use the symbol in a formula where you would normally use a number, you have to interpret the formula differently. Let's look at how does not follow the rules that every actual number does:
Addition Breaks
Every number has a negative, and addition is associative. For we could write and note that . This is a good thing, since it means we can prove if you take one away from infinity, you still have infinity: . But it also means we can prove 1 = 0, which is not so good.
Therefore, .
Reinterpret Formulas that Use
We started off with a formula that does "mean" something, even though it used and is not a number.
What does this mean, compared to what it means when we have a regular number instead of an infinity symbol:
This formula says that I can make sure the values of don't differ very much from , so long as I can control how much x varies away from 2. I don't have to make exactly equal to , but I also can't control x too tightly. I have to give you a range to vary x within. It's just going to be very, very small (probably) if you want to make very very close to . And by the way, it doesn't matter at all what happens when .
If we could use the same paragraph as a template for my original formula, we'll see some problems. Let's substitute 0 for 2, and for .
This formula says that I can make sure the values of don't differ very much from , so long as I can control how much x varies away from 0. I don't have to make exactly equal to , but I also can't control x too tightly. I have to give you a range to vary x within. It's just going to be very, very small (probably) if you want to see that gets very, very close to . And by the way, it doesn't matter at all what happens when .
It's close to making sense, but it isn't quite there. It doesn't make sense to say that some real number is really "close" to . For example, when and does it really makes sense to say 1000 is closer to than 1 is? Solve the following equations for δ:
No real number is very close to ; that's what makes so special! So we have to rephrase the paragraph:
This formula says that I can make sure the values of get as big as any number you pick, so long as I can control how much x varies away from 0. I don't have to make bigger than every number, but I also can't control x too tightly. I have to give you a range to vary x within. It's just going to be very, very small (probably) if you want to see that gets very, very large. And by the way, it doesn't matter at all what happens when .
You can see that the essential nature of the formula hasn't changed, but the exact details require some human interpretation. While rigorous definitions and clear distinctions are essential to the study of mathematics, sometimes a bit of casual rewording is okay. You just have to make sure you understand what a formula really means so you can draw conclusions correctly.
Exercises
Write out an explanatory paragraph for the following limits that include . Remember that you will have to change any comparison of magnitude between a real number and to a different phrase. In the second case, you will have to work out for yourself what the formula means.
<h1> 2.4 Continuity</h1>
Defining Continuity
We are now ready to define the concept of a function being continuous. The idea is that we want to say that a function is continuous if you can draw its graph without taking your pencil off the page. But sometimes this will be true for some parts of a graph but not for others. Therefore, we want to start by defining what it means for a function to be continuous at one point. The definition is simple, now that we have the concept of limits:
If is defined on an open interval containing , then is said to be continuous at if and only if
.Note that for to be continuous at , the definition in effect requires three conditions:
 that is defined at , so exists,
 the limit as approaches exists, and
 the limit and are equal.
If any of these do not hold then is not continuous at .
The idea of the definition is that the point of the graph corresponding to will be close to the points of the graph corresponding to nearby values. Now we can define what it means for a function to be continuous in general, not just at one point.
A function is said to be continuous on if it is continuous at every point of the interval .
We often use the phrase "the function is continuous" to mean that the function is continuous at every real number. This would be the same as saying the function was continuous on (−∞, ∞), but it is a bit more convenient to simply say "continuous".
Note that, by what we already know, the limit of a rational, exponential, trigonometric or logarithmic function at a point is just its value at that point, so long as it's defined there. So, all such functions are continuous wherever they're defined. (Of course, they can't be continuous where they're not defined!)
Discontinuities
A discontinuity is a point where a function is not continuous. There are lots of possible ways this could happen, of course. Here we'll just discuss two simple ways.
Removable discontinuities
The function is not continuous at . It is discontinuous at that point because the fraction then becomes , which is undefined. Therefore the function fails the first of our three conditions for continuity at the point 3; 3 is just not in its domain.
However, we say that this discontinuity is removable. This is because, if we modify the function at that point, we can eliminate the discontinuity and make the function continuous. To see how to make the function continuous, we have to simplify , getting . We can define a new function where . Note that the function is not the same as the original function , because is defined at , while is not. Thus, is continuous at , since . However, whenever , ; all we did to to get was to make it defined at .
In fact, this kind of simplification is often possible with a discontinuity in a rational function. We can divide the numerator and the denominator by a common factor (in our example ) to get a function which is the same except where that common factor was 0 (in our example at ). This new function will be identical to the old except for being defined at new points where previously we had division by 0.
However, this is not possible in every case. For example, the function has a common factor of in both the numerator and denominator, but when you simplify you are left with , which is still not defined at . In this case the domain of and are the same, and they are equal everywhere they are defined, so they are in fact the same function. The reason that differed from in the first example was because we could take it to have a larger domain and not simply that the formulas defining and were different.
Jump discontinuities
Not all discontinuities can be removed from a function. Consider this function:
Since does not exist, there is no way to redefine at one point so that it will be continuous at 0. These sorts of discontinuities are called nonremovable discontinuities.
Note, however, that both onesided limits exist; and . The problem is that they are not equal, so the graph "jumps" from one side of 0 to the other. In such a case, we say the function has a jump discontinuity. (Note that a jump discontinuity is a kind of nonremovable discontinuity.)
OneSided Continuity
Just as a function can have a onesided limit, a function can be continuous from a particular side. For a function to be continuous at a point from a given side, we need the following three conditions:
 the function is defined at the point,
 the function has a limit from that side at that point and
 the onesided limit equals the value of the function at the point.
A function will be continuous at a point if and only if it is continuous from both sides at that point. Now we can define what it means for a function to be continuous on a closed interval.
A function is said to be continuous on if and only if
 it is continuous on ,
 it is continuous from the right at and
 it is continuous from the left at .
Notice that, if a function is continuous, then it is continuous on every closed interval contained in its domain.
Intermediate Value Theorem
A useful theorem regarding continuous functions is the following:
If a function is continuous on a closed interval , then for every value between and there is a value between and such that .
Application: bisection method
The bisection method is the simplest and most reliable algorithm to find zeros of a continuous function.
Suppose we want to solve the equation . Given two points and such that and have opposite signs, the intermediate value theorem tells us that must have at least one root between and as long as is continuous on the interval . If we know is continuous in general (say, because it's made out of rational, trigonometric, exponential and logarithmic functions), then this will work so long as is defined at all points between and . So, let's divide the interval in two by computing . There are now three possibilities:
 ,
 and have opposite signs, or
 and have opposite signs.
In the first case, we're done. In the second and third cases, we can repeat the process on the subinterval where the sign change occurs. In this way we hone in to a small subinterval containing the zero. The midpoint of that small subinterval is usually taken as a good approximation to the zero.
Note that, unlike the methods you may have learned in algebra, this works for any continuous function that you (or your calculator) know how to compute.
<h1> 2.5 Formal Definition of the Limit</h1>
In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (after all, it took mathematicians 150 years to arrive at it); it is also the most important and most useful.
The intuitive definition of a limit is inadequate to prove anything rigorously about it. The problem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of this term is that the closer gets to the specified value, the closer the function must get to the limit, so that however close we want the function to the limit, we can accomplish this by making sufficiently close to our value. We can express this requirement technically as follows:
Let be a function defined on an open interval that contains , except possibly at . Let be a number. Then we say that
if, for every , there exists a such that for all with
we have
 .
To further explain, earlier we said that "however close we want the function to the limit, we can find a corresponding close to our value." Using our new notation of epsilon () and delta (), we mean that if we want to make within of , the limit, then we know that making within of puts it there.
Again, since this is tricky, let's resume our example from before: , at . To start, let's say we want to be within .01 of the limit. We know by now that the limit should be 4, so we say: for , there is some so that as long as , then
To show this, we can pick any that is bigger than 0, so long as it works. For example, you might pick , because you are absolutely sure that if is within .00000000000001 of 2, then will be within .01 of 4. This works for . But we can't just pick a specific value for , like .01, because we said in our definition "for every ." This means that we need to be able to show an infinite number of s, one for each . We can't list an infinite number of s!
Of course, we know of a very good way to do this; we simply create a function, so that for every , it can give us a . In this case, one definition of that works is (see example 5 in choosing delta for an explanation of how this delta was chosen.)
So, in general, how do you show that tends to as tends to ? Well imagine somebody gave you a small number (e.g., say ). Then you have to find a and show that whenever we have . Now if that person gave you a smaller (say ) then you would have to find another , but this time with 0.03 replaced by 0.002. If you can do this for any choice of then you have shown that tends to as tends to . Of course, the way you would do this in general would be to create a function giving you a for every , just as in the example above.
Formal Definition of the Limit at Infinity
We call the limit of as approaches if for every number there exists a such that whenever we have
When this holds we write
or
 as
Similarly, we call the limit of as approaches if for every number , there exists a number such that whenever we have
When this holds we write
or
 as
Notice the difference in these two definitions. For the limit of as approaches we are interested in those such that . For the limit of as approaches we are interested in those such that .
Examples
Here are some examples of the formal definition.
Example 1
We know from earlier in the chapter that
 .
What is when for this limit?
We start with the desired conclusion and substitute the given values for and :
 .
Then we solve the inequality for :
This is the same as saying
 .
(We want the thing in the middle of the inequality to be because that's where we're taking the limit.) We normally choose the smaller of and for , so , but any smaller number will also work.
Example 2
What is the limit of as approaches 4?
There are two steps to answering such a question; first we must determine the answer — this is where intuition and guessing is useful, as well as the informal definition of a limit — and then we must prove that the answer is right.
In this case, 11 is the limit because we know is a continuous function whose domain is all real numbers. Thus, we can find the limit by just substituting 4 in for , so the answer is .
We're not done, though, because we never proved any of the limit laws rigorously; we just stated them. In fact, we couldn't have proved them, because we didn't have the formal definition of the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove that no matter what value of is given to us, we can find a value of such that
whenever
For this particular problem, letting works (see choosing delta for help in determining the value of to use in other problems). Now, we have to prove
given that
 .
Since , we know
which is what we wished to prove.
Example 3
What is the limit of as approaches 4?
As before, we use what we learned earlier in this chapter to guess that the limit is . Also as before, we pull out of thin air that
 .
Note that, since is always positive, so is , as required. Now, we have to prove
given that
 .
We know that
(because of the triangle inequality), so
Example 4
Show that the limit of as approaches 0 does not exist.
We will proceed by contradiction. Suppose the limit exists; call it . For simplicity, we'll assume that ; the case for is similar. Choose . Then if the limit were there would be some such that for every with . But, for every , there exists some (possibly very large) such that , but , a contradiction.
Example 5
What is the limit of as approaches 0?
By the Squeeze Theorem, we know the answer should be 0. To prove this, we let . Then for all , if , then as required.
Example 6
Suppose that and . What is ?
Of course, we know the answer should be , but now we can prove this rigorously. Given some , we know there's a such that, for any with , (since the definition of limit says "for any ", so it must be true for as well). Similarly, there's a such that, for any with , . We can set to be the lesser of and . Then, for any with , , as required.
If you like, you can prove the other limit rules too using the new definition. Mathematicians have already done this, which is how we know the rules work. Therefore, when computing a limit from now on, we can go back to just using the rules and still be confident that our limit is correct according to the rigorous definition.
Formal Definition of a Limit Being Infinity
Let be a function defined on an open interval that contains , except possibly at . Then we say that
if, for every , there exists a such that for all with
we have
 .
When this holds we write
or
 as .
Similarly, we say that
if, for every , there exists a such that for all with
we have
 .
When this holds we write
or
 as .
<h1> 2.6 Proofs of Some Basic Limit Rules</h1>
Now that we have the formal definition of a limit, we can set about proving some of the properties we stated earlier in this chapter about limits.
Constant Rule for Limits
 If b and c are constants then .
Proof of the Constant Rule for Limits:
To prove that , we need to find a such that for every , whenever . and , so is satisfied independent of any value of ; that is, we can choose any we like and the condition holds.
Identity Rule for Limits
 If c is a constant then .
Proof of the Identity Rule for Limits:
To prove that , we need to find a such that for every , whenever . Choosing satisfies this condition.
Scalar Product Rule for Limits
Proof of the Scalar Product Rule for Limits:
Since we are given that , there must be some function, call it , such that for every , whenever . Now we need to find a such that for all , whenever .
First let's suppose that . , so . In this case, letting satisfies the limit condition.
Now suppose that . Since has a limit at , we know from the definition of a limit that is defined in an open interval D that contains (except maybe at itself). In particular, we know that doesn't blow up to infinity within D (except maybe at , but that won't affect the limit), so that in D. Since is the constant function in D, the limit by the Constant Rule for Limits.
Finally, suppose that . , so . In this case, letting satisfies the limit condition.
Sum Rule for Limits
Suppose that and . Then
Proof of the Sum Rule for Limits:
Since we are given that and , there must be functions, call them and , such that for all , whenever , and . whenever .
Adding the two inequalities gives . By the triangle inequality we have , so we have whenever and . Let be the smaller of and . Then this satisfies the definition of a limit for having limit .
Difference Rule for Limits
Suppose that and . Then
Proof of the Difference Rule for Limits: Define . By the Scalar Product Rule for Limits, . Then by the Sum Rule for Limits, .
Product Rule for Limits
Suppose that and . Then
Proof of the Product Rule for Limits:^{[1]}
Let be any positive number. The assumptions imply the existence of the positive numbers such that
 when
 when
 when
According to the condition (3) we see that
 when
Supposing then that and using (1) and (2) we obtain
Quotient Rule for Limits
Suppose that and and . Then
Proof of the Quotient Rule for Limits:
If we can show that , then we can define a function, as and appeal to the Product Rule for Limits to prove the theorem. So we just need to prove that .
Let be any positive number. The assumptions imply the existence of the positive numbers such that
 when
 when
According to the condition (2) we see that
 when
which implies that
 when
Supposing then that and using (1) and (3) we obtain
Suppose that holds for all in some open interval containing , except possibly at itself. Suppose also that . Then also.
Proof of the Squeeze Theorem:
From the assumptions, we know that there exists a such that and when .
These inequalities are equivalent to and when .
Using what we know about the relative ordering of , and , we have
when .
or
when .
So
when .
Notes
 ↑ This proof is adapted from one found at planetmath.org/encyclopedia/ProofOfLimitRuleOfProduct.html due to Planet Math user pahio and made available under the terms of the Creative Commons By/ShareAlike License.
<h1> 2.7 Limits Cumulative Exercises</h1>
Basic Limit Exercises
OneSided Limits
Evaluate the following limits or state that the limit does not exist.
TwoSided Limits
Evaluate the following limits or state that the limit does not exist.
Limits to Infinity
Evaluate the following limits or state that the limit does not exist.
Limits of Piecewise Functions
Evaluate the following limits or state that the limit does not exist.
37. Consider the function
38. Consider the function
39. Consider the function
External Links
Differentiation
Basics of Differentiation
<h1> 3.1 Differentiation Defined</h1>
What is Differentiation?
Differentiation is an operation that allows us to find a function that outputs the rate of change of one variable with respect to another variable.
Informally, we may suppose that we're tracking the position of a car on a twolane road with no passing lanes. Assuming the car never pulls off the road, we can abstractly study the car's position by assigning it a variable, . Since the car's position changes as the time changes, we say that is dependent on time, or . This tells where the car is at each specific time. Differentiation gives us a function which represents the car's speed, that is the rate of change of its position with respect to time.
Equivalently, differentiation gives us the slope at any point of the graph of a nonlinear function. For a linear function, of form , is the slope. For nonlinear functions, such as , the slope can depend on ; differentiation gives us a function which represents this slope.
The Definition of Slope
Historically, the primary motivation for the study of differentiation was the tangent line problem: for a given curve, find the slope of the straight line that is tangent to the curve at a given point. The word tangent comes from the Latin word tangens, which means touching. Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope. But what exactly do we mean by "slope" for a curve?
The solution is obvious in some cases: for example, a line is its own tangent; the slope at any point is . For the parabola , the slope at the point is ; the tangent line is horizontal.
But how can you find the slope of, say, at ? This is in general a nontrivial question, but first we will deal carefully with the slope of lines.
Of a line
The slope of a line, also called the gradient of the line, is a measure of its inclination. A line that is horizontal has slope 0, a line from the bottom left to the top right has a positive slope and a line from the top left to the bottom right has a negative slope.
The slope can be defined in two (equivalent) ways. The first way is to express it as how much the line climbs for a given motion horizontally. We denote a change in a quantity using the symbol (pronounced "delta"). Thus, a change in is written as . We can therefore write this definition of slope as:
An example may make this definition clearer. If we have two points on a line, and , the change in from to is given by:
Likewise, the change in from to is given by:
This leads to the very important result below.
The slope of the line between the points and is

 .
Alternatively, we can define slope trigonometrically , using the tangent function:
where is the angle from the rightwardpointing horizontal to the line, measured counterclockwise. If you recall that the tangent of an angle is the ratio of the ycoordinate to the xcoordinate on the unit circle, you should be able to spot the equivalence here.
Of a graph of a function
The graphs of most functions we are interested in are not straight lines (although they can be), but rather curves. We cannot define the slope of a curve in the same way as we can for a line. In order for us to understand how to find the slope of a curve at a point, we will first have to cover the idea of tangency. Intuitively, a tangent is a line which just touches a curve at a point, such that the angle between them at that point is zero. Consider the following four curves and lines:
(i)  (ii) 
(iii)  (iv) 
 The line crosses, but is not tangent to at .
 The line crosses, and is tangent to at .
 The line crosses at two points, but is tangent to only at .
 There are many lines that cross at , but none are tangent. In fact, this curve has no tangent at .
A secant is a line drawn through two points on a curve. We can construct a definition of a tangent as the limit of a secant of the curve taken as the separation between the points tends to zero. Consider the diagram below.
As the distance tends to zero, the secant line becomes the tangent at the point . The two points we draw our line through are:
and
As a secant line is simply a line and we know two points on it, we can find its slope, , using the formula from before:
(We will refer to the slope as because it may, and generally will, depend on .) Substituting in the points on the line,
This simplifies to
This expression is called the difference quotient. Note that can be positive or negative — it is perfectly valid to take a secant through any two points on the curve — but cannot be .
The definition of the tangent line we gave was not rigorous, since we've only defined limits of numbers — or, more precisely, of functions that output numbers — not of lines. But we can define the slope of the tangent line at a point rigorously, by taking the limit of the slopes of the secant lines from the last paragraph. Having done so, we can then define the tangent line as well. Note that we cannot simply set to zero as this would imply division of zero by zero which would yield an undefined result. Instead we must find the limit of the above expression as tends to zero:
The slope of the graph of at the point is
If this limit does not exist, then we say the slope is undefined.
If the slope is defined, say , then the tangent line to the graph of at the point is the line with equation
This last equation is just the pointslope form for the line through with slope .
Exercises
The Rate of Change of a Function at a Point
Consider the formula for average velocity in the direction, , where is the change in over the time interval . This formula gives the average velocity over a period of time, but suppose we want to define the instantaneous velocity. To this end we look at the change in position as the change in time approaches 0. Mathematically this is written as: , which we abbreviate by the symbol . (The idea of this notation is that the letter denotes change.) Compare the symbol with . The idea is that both indicate a difference between two numbers, but denotes a finite difference while denotes an infinitesimal difference. Please note that the symbols and have no rigorous meaning on their own, since , and we can't divide by 0.
(Note that the letter is often used to denote distance, which would yield . The letter is often avoided in denoting distance due to the potential confusion resulting from the expression .)
The Definition of the Derivative
You may have noticed that the two operations we've discussed — computing the slope of the tangent to the graph of a function and computing the instantaneous rate of change of the function — involved exactly the same limit. That is, the slope of the tangent to the graph of is . Of course, can, and generally will, depend on , so we should really think of it as a function of . We call this process (of computing ) differentiation. Differentiation results in another function whose value for any value is the slope of the original function at . This function is known as the derivative of the original function.
Since lots of different sorts of people use derivatives, there are lots of different mathematical notations for them. Here are some:
 (read "f prime of x") for the derivative of ,
 ,
 ,
 for the derivative of as a function of or
 , which is more useful in some cases.
Most of the time the brackets are not needed, but are useful for clarity if we are dealing with something like , where we want to differentiate the product of two functions, and .
The first notation has the advantage that it makes clear that the derivative is a function. That is, if we want to talk about the derivative of at , we can just write .
In any event, here is the formal definition:
Let be a function. Then wherever this limit exists. In this case we say that is differentiable at and its derivative at is .
Examples
Example 1
The derivative of is
 ,
no matter what is. This is consistent with the definition of the derivative as the slope of a function.
Example 2
What is the slope of the graph of at ? We can do it "the hard (and imprecise) way", without using differentiation, as follows, using a calculator and using small differences below and above the given point:
When , .
When , .
Then the difference between the two values of is .
Then the difference between the two values of is .
Thus, the slope at the point of the graph at which .
But, to solve the problem precisely, we compute

= = = = = = .
We were lucky this time; the approximation we got above turned out to be exactly right. But this won't always be so, and, anyway, this way we didn't need a calculator.
In general, the derivative of is

= = = = = = = .
Example 3
If (the absolute value function) then , which can also be stated as Finding this derivative is a bit complicated, so we won't prove it at this point.
Here, is not smooth (though it is continuous) at and so the limits and (the limits as 0 is approached from the right and left respectively) are not equal. From the definition, , which does not exist. Thus, is undefined, and so has a discontinuity at 0. This sort of point of nondifferentiability is called a cusp. Functions may also not be differentiable because they go to infinity at a point, or oscillate infinitely frequently.
Understanding the derivative notation
The derivative notation is special and unique in mathematics. The most common notation for derivatives you'll run into when first starting out with differentiating is the Leibniz notation, expressed as . You may think of this as "rate of change in with respect to ". You may also think of it as "infinitesimal value of divided by infinitesimal value of ". Either way is a good way of thinking, although you should remember that the precise definition is the one we gave above. Often, in an equation, you will see just , which literally means "derivative with respect to x". This means we should take the derivative of whatever is written to the right; that is, means where .
As you advance through your studies, you will see that we sometimes pretend that and are separate entities that can be multiplied and divided, by writing things like . Eventually you will see derivatives such as , which just means that the input variable of our function is called and our output variable is called ; sometimes, we will write , to mean the derivative with respect to of whatever is written on the right. In general, the variables could be anything, say .
All of the following are equivalent for expressing the derivative of
Exercises
Differentiation Rules
The process of differentiation is tedious for complicated functions. Therefore, rules for differentiating general functions have been developed, and can be proved with a little effort. Once sufficient rules have been proved, it will be fairly easy to differentiate a wide variety of functions. Some of the simplest rules involve the derivative of linear functions.
Derivative of a constant function
For any fixed real number ,
Intuition
The graph of the function is a horizontal line, which has a constant slope of zero. Therefore, it should be expected that the derivative of this function is zero, regardless of the values of and .
Proof
The definition of a derivative is
Let for all . (That is, is a constant function.) Then . Therefore
 .
Let . To prove that , we need to find a positive such that, for any given positive , whenever . But , so for any choice of .
Examples
Note that, in the second example, is just a constant.
Derivative of a linear function
For any fixed real numbers and ,
The special case shows the advantage of the notation—rules are intuitive by basic algebra, though this does not constitute a proof, and can lead to misconceptions to what exactly and actually are.
Intuition
The graph of is a line with constant slope .
Proof
If , then . So,

= = = = =
Constant multiple and addition rules
Since we already know the rules for some very basic functions, we would like to be able to take the derivative of more complex functions by breaking them up into simpler functions. Two tools that let us do this are the constant multiple rule and the addition rule.
The Constant Rule
For any fixed real number ,
The reason, of course, is that one can factor out of the numerator, and then of the entire limit, in the definition. The details are left as an exercise.
Example
We already know that
 .
Suppose we want to find the derivative of

= = =
Another simple rule for breaking up functions is the addition rule.
The Addition and Subtraction Rules
Proof
From the definition:
By definition then, this last term is
Example
What is the derivative of ?

= = = =
The fact that both of these rules work is extremely significant mathematically because it means that differentiation is linear. You can take an equation, break it up into terms, figure out the derivative individually and build the answer back up, and nothing odd will happen.
We now need only one more piece of information before we can take the derivatives of any polynomial.
The Power Rule
This has been proved in an example in Derivatives of Exponential and Logarithm Functions where it can be best understood.
For example, in the case of the derivative is as was established earlier. A special case of this rule is that .
Since polynomials are sums of monomials, using this rule and the addition rule lets you differentiate any polynomial. A relatively simple proof for this can be derived from the binomial expansion theorem.
This rule also applies to fractional and negative powers. Therefore

= = =
Derivatives of polynomials
With these rules in hand, you can now find the derivative of any polynomial you come across. Rather than write the general formula, let's go step by step through the process.
The first thing we can do is to use the addition rule to split the equation up into terms:
We can immediately use the linear and constant rules to get rid of some terms:
Now you may use the constant multiplier rule to move the constants outside the derivatives:
Then use the power rule to work with the individual monomials:
And then do some algebra to get the final answer:
These are not the only differentiation rules. There are other, more advanced, differentiation rules, which will be described in a later chapter.
Exercises
 Find the derivatives of the following equations:
<h1> 3.2 Product and Quotient Rules</h1>
Product Rule
When we wish to differentiate a more complicated expression such as
our only way (up to this point) to differentiate the expression is to expand it and get a polynomial, and then differentiate that polynomial. This method becomes very complicated and is particularly error prone when doing calculations by hand. A beginner might guess that the derivative of a product is the product of the derivatives, similar to the sum and difference rules, but this is not true. To take the derivative of a product, we use the product rule.

It may also be stated as
or in the Leibniz notation as
 .
The derivative of the product of three functions is:
 .
Since the product of two or more functions occurs in many mathematical models of physical phenomena, the product rule has broad application in physics, chemistry, and engineering.
Examples
 Suppose one wants to differentiate ƒ(x) = x^{2} sin(x). By using the product rule, one gets the derivative ƒ '(x) = 2x sin(x) + x^{2}cos(x) (since the derivative of x^{2} is 2x and the derivative of sin(x) is cos(x)).
 One special case of the product rule is the constant multiple rule, which states: if c is a real number and ƒ(x) is a differentiable function, then cƒ(x) is also differentiable, and its derivative is (c × ƒ)'(x) = c × ƒ '(x). This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear.
Physics Example I: rocket acceleration
Consider the vertical acceleration of a model rocket relative to its initial position at a fixed point on the ground. Newton's second law says that the force is equal to the time rate change of momentum. If F is the net force (sum of forces), p is the momentum, and t is the time,
Since the momentum is equal to the product of mass and velocity, this yields
where m is the mass and v is the velocity. Application of the product rule gives
Since the acceleration, a, is defined as the time rate change of velocity, a = dv/dt,
Solving for the acceleration,
Since the rocket is losing mass, dm/dt is negative, and the changing mass term results in increased acceleration.^{[1]}^{[2]}
Note: Here is considered to be the net force.
Physics Example II: electromagnetic induction
Faraday's law of electromagnetic induction states that the induced electromotive force is the negative time rate of change of magnetic flux through a conducting loop.
where is the electromotive force (emf) in volts and Φ_{B} is the magnetic flux in webers. For a loop of area, A, in a magnetic field, B, the magnetic flux is given by
where θ is the angle between the normal to the current loop and the magnetic field direction.
Taking the negative derivative of the flux with respect to time yields the electromotive force gives
In many cases of practical interest only one variable (A, B, or θ) is changing, so two of the three above terms are often zero.
Physics Example III: Kinematics
The position of a particle on a number line relative to a fixed point O is , where represents the time. What is its instantaneous velocity at relative to O? Distances are in meters and time in seconds.
Answer:
Note: To solve this problem, we need some 'tools' from the next section.
We can simplify the function to ()
Substituting into our velocity function:
m/s (to 2 decimal places).
Proof of the Product Rule
Proving this rule is relatively straightforward, first let us state the equation for the derivative:
We will then apply one of the oldest tricks in the book—adding a term that cancels itself out to the middle:
Notice that those terms sum to zero, and so all we have done is add 0 to the equation. Now we can split the equation up into forms that we already know how to solve:
Looking at this, we see that we can factor the common terms out of the numerators to get:
Which, when we take the limit, becomes:
 , or the mnemonic "one Dtwo plus two Done"
This can be extended to 3 functions:
For any number of functions, the derivative of their product is the sum, for each function, of its derivative times each other function.
Back to our original example of a product, , we find the derivative by the product rule is
Note, its derivative would not be
which is what you would get if you assumed the derivative of a product is the product of the derivatives.
To apply the product rule we multiply the first function by the derivative of the second and add to that the derivative of first function multiply by the second function. Sometimes it helps to remember the memorize the phrase "First times the derivative of the second plus the second times the derivative of the first."
Quotient Rule
There is a similar rule for quotients. To prove it, we go to the definition of the derivative:
This leads us to the socalled "quotient rule":

Some people remember this rule with the mnemonic "low Dhigh minus high Dlow, square the bottom and away we go!"
Examples
The derivative of is:
Remember: the derivative of a product/quotient is not the product/quotient of the derivatives. (That is, differentiation does not distribute over multiplication or division.) However one can distribute before taking the derivative. That is
References
<h1> 3.3 Derivatives of Trigonometric Functions</h1>
Sine, cosine, tangent, cosecant, secant, cotangent. These are functions that crop up continuously in mathematics and engineering and have a lot of practical applications. They also appear in more advanced mathematics, particularly when dealing with things such as line integrals with complex numbers and alternate representations of space like spherical and cylindrical coordinate systems.
We use the definition of the derivative, i.e.,
 ,
to work these first two out.
Let us find the derivative of sin x, using the above definition.
Definition of derivative  
trigonometric identity  
factoring  
separation of terms  
application of limit  
solution 
Now for the case of cos x.
Definition of derivative  
trigonometric identity  
factoring  
separation of terms  
application of limit  
solution 
Therefore we have established

To find the derivative of the tangent, we just remember that:
which is a quotient. Applying the quotient rule, we get:
Then, remembering that , we simplify:

For secants, we again apply the quotient rule.
Leaving us with:
Simplifying, we get:

Using the same procedure on cosecants:
We get:

Using the same procedure for the cotangent that we used for the tangent, we get:

<h1> 3.4 Chain Rule</h1>
The chain rule is a method to compute the derivative of the functional composition of two or more functions.
If a function, f, depends on a variable, u, which in turn depends on another variable, x, that is f = y(u(x)) , then the rate of change of f with respect to x can be computed as the rate of change of y with respect to u multiplied by the rate of change of u with respect to x.
If a function f is composed to two differentiable functions y(x) and u(x), so that f(x) = y(u(x)), then f(x) is differentiable and, 
The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another.^{[1]} For example, if f is a function of g which is in turn a function of h, which is in turn a function of x, that is
,
the derivative of f with respect to x is given by
and so on.
A useful mnemonic is to think of the differentials as individual entities that can be canceled algebraically, such as
However, keep in mind that this trick comes about through a clever choice of notation rather than through actual algebraic cancellation.
The chain rule has broad applications in physics, chemistry, and engineering, as well as being used to study related rates in many disciplines. The chain rule can also be generalized to multiple variables in cases where the nested functions depend on more than one variable.
Examples
Example I
Suppose that a mountain climber ascends at a rate of 0.5 kilometer per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometer. To calculate the decrease in air temperature per hour that the climber experiences, one multiplies 6 °C per kilometer by 0.5 kilometer per hour, to obtain 3 °C per hour. This calculation is a typical chain rule application.
Example II
Consider the function f(x) = (x^{2} + 1)^{3}. It follows from the chain rule that
Function to differentiate  
Define u(x) as inside function  
Express f(x) in terms of u(x)  
Express chain rule applicable here  
Substitute in f(u) and u(x)  
Compute derivatives with power rule  
Substitute u(x) back in terms of x  
Simplify. 
Example III
In order to differentiate the trigonometric function
one can write:
Function to differentiate  
Define u(x) as inside function  
Express f(x) in terms of u(x)  
Express chain rule applicable here  
Substitute in f(u) and u(x)  
Evaluate derivatives  
Substitute u in terms of x. 
Example IV: absolute value
The chain rule can be used to differentiate , the absolute value function:
Function to differentiate  
Equivalent function  
Define u(x) as inside function  
Express f(x) in terms of u(x)  
Express chain rule applicable here  
Substitute in f(u) and u(x)  
Compute derivatives with power rule  
Substitute u(x) back in terms of x  
Simplify  
Express as absolute value. 
Example V: three nested functions
The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another. For example, if , sequential application of the chain rule yields the derivative as follows (we make use of the fact that , which will be proved in a later section):
Original (outermost) function  
Define h(x) as innermost function  
g(h) = sin(h) as middle function  
Express chain rule applicable here  
Differentiate f(g)^{[2]}  
Differentiate g(h)  
Differentiate h(x)  
Substitute into chain rule. 
Chain Rule in Physics
Because one physical quantity often depends on another, which, in turn depends on others, the chain rule has broad applications in physics. This section presents examples of the chain rule in kinematics and simple harmonic motion. The chain rule is also useful in electromagnetic induction.
Physics Example I: relative kinematics of two vehicles
For example, one can consider the kinematics problem where one vehicle is heading west toward an intersection at 80 miles per hour while another is heading north away from the intersection at 60 miles per hour. One can ask whether the vehicles are getting closer or further apart and at what rate at the moment when the northbound vehicle is 3 miles north of the intersection and the westbound vehicle is 4 miles east of the intersection.
Big idea: use chain rule to compute rate of change of distance between two vehicles.
Plan:
 Choose coordinate system
 Identify variables
 Draw picture
 Big idea: use chain rule to compute rate of change of distance between two vehicles
 Express c in terms of x and y via Pythagorean theorem
 Express dc/dt using chain rule in terms of dx/dt and dy/dt
 Substitute in x, y, dx/dt, dy/dt
 Simplify.
Choose coordinate system: Let the yaxis point north and the xaxis point east.
Identify variables: Define y(t) to be the distance of the vehicle heading north from the origin and x(t) to be the distance of the vehicle heading west from the origin.
Express c in terms of x and y via Pythagorean theorem:
Express dc/dt using chain rule in terms of dx/dt and dy/dt:
Apply derivative operator to entire function  
Sum of squares is inside function  
Distribute differentiation operator  
Apply chain rule to x(t) and y(t)}  
Simplify. 
Substitute in x = 4 mi, y = 3 mi, dx/dt = −80 mi/hr, dy/dt = 60 mi/hr and simplify
Consequently, the two vehicles are getting closer together at a rate of 28 mi/hr.
Physics Example II: harmonic oscillator
If the displacement of a simple harmonic oscillator from equilibrium is given by x, and it is released from its maximum displacement A at time t = 0, then the position at later times is given by
where ω = 2 π/T is the angular frequency and T is the period of oscillation. The velocity, v, being the first time derivative of the position can be computed with the chain rule:
Definition of velocity in one dimension  
Substitute x(t)  
Bring constant A outside of derivative  
Differentiate outside function (cosine)  
Bring negative sign in front  
Evaluate remaining derivative  
Simplify. 
The acceleration is then the second time derivative of position, or simply dv/dt.
Definition of acceleration in one dimension  
Substitute v(t)  
Bring constant term outside of derivative  
Differentiate outside function (sine)  
Evaluate remaining derivative  
Simplify. 
From Newton's second law, F = ma, where F is the net force and m is the object's mass.
Newton's second law  
Substitute a(t)  
Simplify  
Substitute original x(t). 
Thus it can be seen that these results are consistent with the observation that the force on a simple harmonic oscillator is a negative constant times the displacement.
Chain Rule in Chemistry
The chain rule has many applications in Chemistry because many equations in Chemistry describe how one physical quantity depends on another, which in turn depends on another. For example, the ideal gas law describes the relationship between pressure, volume, temperature, and number of moles, all of which can also depend on time.
Chemistry Example I: Ideal Gas Law
Suppose a sample of n moles of an ideal gas is held in an isothermal (constant temperature, T) chamber with initial volume V_{0}. The ideal gas is compressed by a piston so that its volume changes at a constant rate so that V(t) = V_{0}  kt, where t is the time. The chain rule can be employed to find the time rate of change of the pressure.^{[3]} The ideal gas law can be solved for the pressure, P to give:
where P(t) and V(t) have been written as explicit functions of time and the other symbols are constant. Differentiating both sides yields
where the constant terms, n, R, and T, have been moved to the left of the derivative operator. Applying the chain rule gives
where the power rule has been used to differentiate 1/V, Since V(t) = V_{0}  kt, dV/dt = k. Substituting in for V and dV/dt yields dP/dt.
Chemistry Example II: Kinetic Theory of Gases
A second application of the chain rule in Chemistry is finding the rate of change of the average molecular speed, v, in an ideal gas as the absolute temperature T, increases at a constant rate so that T = T_{0} + at, where T_{0} is the initial temperature and t is the time.^{[3]} The kinetic theory of gases relates the root mean square of the molecular speed to the temperature, so that if v(t) and T(t) are functions of time,
where R is the ideal gas constant, and M is the molecular weight.
Differentiating both sides with respect to time yields:
Using the chain rule to express the right side in terms of the with respect to temperature, T, and time, t, respectively gives
Evaluating the derivative with respect to temperature, T, yields
Evaluating the remaining derivative with respect to T, taking the reciprocal of the negative power, and substituting T = T_{0} + at, produces
Evaluating the derivative with respect to t yields
which simplifies to
Proof of the chain rule
Suppose is a function of which is a function of (it is assumed that is differentiable at and , and is differentiable at .
To prove the chain rule we use the definition of the derivative.
We now multiply by and perform some algebraic manipulation.
Note that as approaches 0, also approaches 0. So taking the limit as of a function as approaches 0 is the same as taking its limit as approaches 0. Thus
So we have
Exercises
References
 ↑ http://www.math.brown.edu/help/derivtips.html
 ↑ The derivative of is ; see Calculus/Derivatives of Exponential and Logarithm Functions.
 ↑ ^{a} ^{b} University of British Columbia, UBC Calculus Online Course Notes, Applications of the Chain Rule, http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/chainap.html Accessed 11/15/2010.
External links
<h1> 3.5 Higher Order Derivatives</h1>
The second derivative, or second order derivative, is the derivative of the derivative of a function. The derivative of the function may be denoted by , and its double (or "second") derivative is denoted by . This is read as "f double prime of x," or "The second derivative of f(x)." Because the derivative of function is defined as a function representing the slope of function , the double derivative is the function representing the slope of the first derivative function.
Furthermore, the third derivative is the derivative of the derivative of the derivative of a function, which can be represented by . This is read as "f triple prime of x", or "The third derivative of f(x)". This can continue as long as the resulting derivative is itself differentiable, with the fourth derivative, the fifth derivative, and so on. Any derivative beyond the first derivative can be referred to as a higher order derivative.
Notation
Let be a function in terms of x. The following are notations for higher order derivatives.
2nd Derivative  3rd Derivative  4th Derivative  nth Derivative  Notes 

Probably the most common notation.  
Leibniz notation.  
Another form of Leibniz notation.  
Euler's notation. 
Warning: You should not write to indicate the nth derivative, as this is easily confused with the quantity all raised to the nth power.
The Leibniz notation, which is useful because of its precision, follows from
 .
Newton's dot notation extends to the second derivative, , but typically no further in the applications where this notation is common.
Examples
Example 1:
Find the third derivative of with respect to x.
Repeatedly apply the Power Rule to find the derivatives.
Example 2:
Find the third derivative of with respect to x.
Applications:
For applications of the second derivative in finding a curve's concavity and points of inflection, see "Extrema and Points of Inflection" and "Extreme Value Theorem". For applications of higher order derivatives in physics, see the "Kinematics" section.
<h1> Failed to match page to section number. Check your argument; if correct, consider updating Template:Calculus/map page. Implicit Differentiation</h1>
Generally, you will encounter functions expressed in explicit form, that is, in the form . To find the derivative of y with respect to x, you take the derivative with respect to x of both sides of the equation to get
But suppose you have a relation of the form . In this case, it may be inconvenient or even impossible to solve for y as a function of x. A good example is the relation . In this case you can utilize implicit differentiation to find the derivative. To do so, one takes the derivative of both sides of the equation with respect to x and solves for . That is, form
and solve for dy/dx. You need to employ the chain rule whenever you take the derivative of a variable with respect to a different variable. For example,
Implicit Differentiation and the Chain Rule
To understand how implicit differentiation works and use it effectively it is important to recognize that the key idea is simply the chain rule. First let's recall the chain rule. Suppose we are given two differentiable functions f(x) and g(x) and that we are interested in computing the derivative of the function f(g(x)), the the chain rule states that:
That is, we take the derivative of f as normal and then plug in g, finally multiply the result by the derivative of g.
Now suppose we want to differentiate a term like y^{2} with respect to x where we are thinking of y as a function of x, so for the remainder of this calculation let's write it as y(x) instead of just y. The term y^{2} is just the composition of f(x) = x^{2} and y(x). That is, f(y(x)) = y^{2}(x). Recalling that f′(x) = 2x then the chain rule states that:
Of course it is customary to think of y as being a function of x without always writing y(x), so this calculation usually is just written as
Don't be confused by the fact that we don't yet know what y′ is, it is some function and often if we are differentiating two quantities that are equal it becomes possible to explicitly solve for y′ (as we will see in the examples below.) This makes it a very powerful technique for taking derivatives.
Explicit Differentiation
For example, suppose we are interested in the derivative of y with respect to x where x and y are related by the equation
This equation represents a circle of radius 1 centered on the origin. Note that y is not a function of x since it fails the vertical line test ( when , for example).
To find y', first we can separate variables to get
Taking the square root of both sides we get two separate functions for y:
We can rewrite this as a fractional power:
Using the chain rule we get,
And simplifying by substituting y back into this equation gives
Implicit Differentiation
Using the same equation
First, differentiate with respect to x on both sides of the equation:
To differentiate the second term on the left hand side of the equation (call it f(y(x))=y^{2}), use the chain rule:
So the equation becomes
Separate the variables:
Divide both sides by , and simplify to get the same result as above:
Uses
Implicit differentiation is useful when differentiating an equation that cannot be explicitly differentiated because it is impossible to isolate variables.
For example, consider the equation,
Differentiate both sides of the equation (remember to use the product rule on the term xy):
Isolate terms with y':
Factor out a y' and divide both sides by the other term:
Example
can be solved as:
then differentiated:
However, using implicit differentiation it can also be differentiated like this:
use the product rule:
solve for :
Note that, if we substitute into , we end up with again.
Application: inverse trigonometric functions
Arcsine, arccosine, arctangent. These are the functions that allow you to determine the angle given the sine, cosine, or tangent of that angle.
First, let us start with the arcsine such that:
To find dy/dx we first need to break this down into a form we can work with:
Then we can take the derivative of that:
...and solve for dy / dx:
At this point we need to go back to the unit triangle. Since y is the angle and the opposite side is sin(y) (which is equal to x), the adjacent side is cos(y) (which is equal to the square root of 1 minus x^{2}, based on the Pythagorean theorem), and the hypotenuse is 1. Since we have determined the value of cos(y) based on the unit triangle, we can substitute it back in to the above equation and get:

We can use an identical procedure for the arccosine and arctangent:


<h1> 3.7 Derivatives of Exponential and Logarithm Functions</h1>
Logarithm Function
We shall first look at the value of :
Now we find the derivative of using the formal definition of the derivative:
Let . Note that as approaches , approaches 0. So we can redefine our limit as:
Here we could take the natural logarithm outside the limit because it doesn't have anything to do with the limit (we could have chosen not to). We then substituted the value of .

If we wanted, we could go through that same process again for a generalized base, but it is easier just to use properties of logs and realize that:
Since 1 / ln(b) is a constant, we can just take it outside of the derivative:
Which leaves us with the generalized form of:

Exponential Function
We shall take two different approaches to finding the derivative of . The first approach:
The second approach:
Note that in the second approach we made some use of the chain rule.Thus:
so that we have proved the following rule:

Now that we have derived a specific case, let us extend things to the general case. Assuming that a is a positive real constant, we wish to calculate:
One of the oldest tricks in mathematics is to break a problem down into a form that we already know we can handle. Since we have already determined the derivative of e^{x}, we will attempt to rewrite a^{x} in that form.
Using that e^{ln(c)} = c and that ln(a^{b}) = b · ln(a), we find that:
Thus, we simply apply the chain rule:

Logarithmic Differentiation
We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such a products with many terms, quotients of composed functions, or functions with variable or function exponents. We do this by taking the natural logarithm of both sides, rearranging terms using the logarithm laws below, and then differentiating both sides implicitly, before multiplying through by y.

See the examples below.
Example 1
We shall now prove the validity of the power rule using logarithmic differentiation.
Thus:
 Example 2
Suppose we wished to differentiate We take the natural logarithm of both sides Differentiating implicitly, recalling the chain rule Multiplying by y, the original function 
 Example 3
Let us differentiate a function Taking the natural logarithm of left and right We then differentiate both sides, recalling the product and chain rules Multiplying by the original function y 
 Example 4
Take a function Then We then differentiate And finally multiply by y 
<h1> 3.8 Some Important Theorems</h1>
This section covers three theorems of fundamental importance to the topic of differential calculus: The Extreme Value Theorem, Rolle's Theorem, and the Mean Value Theorem. It also discusses the relationship between differentiability and continuity.
Extreme Value Theorem
Classification of Extrema
We start out with some definitions.
A global maximum (also called an absolute maximum) of a function on a closed interval is a value such that for all in .
A global minimum (also called an absolute minimum) of a function on a closed interval is a value such that for all in .
Maxima and minima are collectively known as extrema.
The Extreme Value Theorem
If is a function that is continuous on the closed interval [], then has both a global minimum and a global maximum on []. It is assumed that a and b are both finite.
The Extreme Value Theorem is a fundamental result of real analysis whose proof is beyond the scope of this text. However, the truth of the theorem allows us to talk about the maxima and minima of continuous functions on closed intervals without concerning ourselves with whether or not they exist. When dealing with functions that do not satisfy the premises of the theorem, we will need to worry about such things. For example, the unbounded function has no extrema whatsoever. If is restricted to the semiclosed interval [), then has a minimum value of at , but it has no maximum value since, for any given value in , one can always find a larger value of for in , for example by forming , where is the average of with . The function has a discontinuity at . fails to have any extrema in any closed interval around since the function is unbounded below as one approaches from the left, and it is unbounded above as one approaches from the right. (In fact, the function is undefined for x=0. However, the example is unaffected if g(0) is assigned any arbitrary value.)
The Extreme Value Theorem is an existence theorem. It tells us that global extrema exist if certain conditions are met, but it doesn't tell us how to find them. We will discuss how to determine the extrema of continuous functions in the section titled Extrema and Points of Inflection.
Rolle's Theorem
If a function, , is continuous on the closed interval , is differentiable on the open interval , and , then there exists at least one number c, in the interval such that
Rolle's Theorem is important in proving the Mean Value Theorem. Intuitively it says that if you have a function that is continuous everywhere in an interval bounded by points where the function has the same value, and if the function is differentiable everywhere in the interval (except maybe at the endpoints themselves), then the function must have zero slope in at least one place in the interior of the interval.
Proof of Rolle's Theorem
If is constant on , then for every in , so the theorem is true. So for the remainder of the discussion we assume is not constant on .
Since satisfies the conditions of the Extreme Value Theorem, must attain its maximum and minimum values on . Since is not constant on , the endpoints cannot be both maxima and minima. Thus, at least one extremum exists in . We can suppose without loss of generality that this extremum is a maximum because, if it were a minimum, we could consider the function instead. Let with in be a maximum. It remains to be shown that .
By the definition of derivative, . By substituting , this is equivalent to . Note that for all in since is the maximum on .
since it has nonpositive numerator and negative denominator.
since it has nonpositive numerator and positive denominator.
The limits from the left and right must be equal since the function is differentiable at , so .
Exercise
Mean Value Theorem
If is continuous on the closed interval , where ,and differentiable on the open interval , there exists at least one in the open interval such that
 .
The Mean Value Theorem is an important theorem of differential calculus. It basically says that for a differentiable function defined on an interval, there is some point on the interval whose instantaneous slope is equal to the average slope of the interval. Note that Rolle's Theorem is the special case of the Mean Value Theorem when .
In order to prove the Mean Value Theorem, we will prove a more general statement, of which the Mean Value Theorem is a special case. The statement is Cauchy's Mean Value Theorem, also known as the Extended Mean Value Theorem.
Cauchy's Mean Value Theorem
If , are continuous on the closed interval and differentiable on the open interval , then there exists a number, , in the open interval such that
If and , then this is equivalent to
 .
To prove Cauchy's Mean Value Theorem, consider the function . Since both and are continuous on and differentiable on , so is . .Since (see the exercises), Rolle's Theorem tells us that there exists some number in such that . This implies that , which is what was to be shown.
Exercises
Differentiability Implies Continuity
If exists then is continuous at . To see this, note that . But
This imples that or , which shows that is continuous at .
The converse, however, is not true. Take , for example. is continuous at 0 since and and , but it is not differentiable at 0 since but .
<h1> 3.9 Basics of Differentiation Cumulative Exercises</h1>
Find the Derivative by Definition
Find the derivative of the following functions using the limit definition of the derivative.
Prove the Constant Rule
Find the Derivative by Rules
Find the derivative of the following functions:
Power Rule
Product Rule
Quotient Rule
Chain Rule
Exponentials
Logarithms
Trigonometric functions
More Differentiation
Implicit Differentiation
Use implicit differentiation to find y'
Logarithmic Differentiation
Use logarithmic differentiation to find :
Equation of Tangent Line
For each function, , (a) determine for what values of the tangent line to is horizontal and (b) find an equation of the tangent line to at the given point.
Higher Order Derivatives
External Links
Applications of Derivatives
<h1> Failed to match page to section number. Check your argument; if correct, consider updating Template:Calculus/map page. L'Hôpital's Rule</h1>
L'Hopital's Rule
Occasionally, one comes across a limit which results in or , which are called indeterminate limits. However, it is still possible to solve these in many cases due to L'Hôpital's rule. This rule also is vital in explaining how a number of other limits can be derived.
If exists, where or , the limit is said to be indeterminate.
All of the following expressions are indeterminate forms.
These expressions are called indeterminate because you cannot determine their exact value in the indeterminate form. Depending on the situation, each indeterminate form could evaluate to a variety of values.
Theorem
If is indeterminate of type or ,
then
In other words, if the limit of the function is indeterminate, the limit equals the derivative of the top over the derivative of the bottom. If that is indeterminate, L'Hôpital's rule can be used again until the limit isn't or .
Note:
can approach a finite value c, or .
Proof of the 0/0 case
Suppose that for real functions and , and and that exists. Thus and exist in an interval around , but maybe not at itself. This implies that both and are differentiable (and thus continuous) everywhere in except perhaps at . Thus, for any in , in any interval or , and are continuous and differentiable, with the possible exception of . Define
and .
Note that , and that and are continuous in any interval or and differentiable in any interval or when is in . Cauchy's Mean Value Theorem tells us that for some in (if ) or (if ). Since , we have for and in . Note that is the same limit as since both and are being squeezed to . So taking the limit as of the last equation gives which is equivalent to .
Examples
Example 1
Find
Since plugging in 0 for x results in , use L'Hôpital's rule to take the derivative of the top and bottom, giving:
Plugging in 0 for x gives 1 here.
Example 2
Find
First, you need to rewrite the function into an indeterminate limit fraction:
Now it's indeterminate. Take the derivative of the top and bottom:
Plugging in 0 for x once again gives 1.
Example 3
Find
This time, plugging in for x gives you . You know the drill:
This time, though, there is no x term left! is the answer.
Example 4
Sometimes, forms exist where it is not intuitively obvious how to solve them. One might think the value However, as was noted in the definition of an indeterminate form, this isn't possible to evaluate using the rules learned before now, and we need to use L'Hôpital's rule to solve.
Find
Plugging the value of x into the limit yields
 (indeterminate form).
Let

= = = (indeterminate form)
We now apply L'Hôpital's rule by taking the derivative of the top and bottom with respect to x.
Returning to the expression above

= = (indeterminate form)
We apply L'Hôpital's rule once again
Therefore
And
Careful: this does not prove that because
Exercises
Evaluate the following limits using L'Hôpital's rule:
<h1> 3.11 Extrema and Points of Inflection</h1>
Maxima and minima are points where a function reaches a highest or lowest value, respectively. There are two kinds of extrema (a word meaning maximum or minimum): global and local, sometimes referred to as "absolute" and "relative", respectively. A global maximum is a point that takes the largest value on the entire range of the function, while a global minimum is the point that takes the smallest value on the range of the function. On the other hand, local extrema are the largest or smallest values of the function in the immediate vicinity.
In many cases, extrema look like the crest of a hill or the bottom of a bowl on a graph of the function. A global extremum is always a local extremum too, because it is the largest or smallest value on the entire range of the function, and therefore also its vicinity. It is also possible to have a function with no extrema, global or local: y=x is a simple example.
At any extremum, the slope of the graph is necessarily zero (or is undefined, as in the case of x), as the graph must stop rising or falling at an extremum, and begin to head in the opposite direction. Because of this, extrema are also commonly called stationary points or turning points. Therefore, the first derivative of a function is equal to zero at extrema. If the graph has one or more of these stationary points, these may be found by setting the first derivative equal to zero and finding the roots of the resulting equation.
However, a slope of zero does not guarantee a maximum or minimum: there is a third class of stationary point called a saddle point. Consider the function

 .
The derivative is
The slope at x=0 is 0. We have a slope of zero, but while this makes it a stationary point, this doesn't mean that it is a maximum or minimum. Looking at the graph of the function you will see that x=0 is neither, it's just a spot at which the function flattens out. True extrema require a sign change in the first derivative. This makes sense  you have to rise (positive slope) to and fall (negative slope) from a maximum. In between rising and falling, on a smooth curve, there will be a point of zero slope  the maximum. A minimum would exhibit similar properties, just in reverse.
This leads to a simple method to classify a stationary point  plug x values slightly left and right into the derivative of the function. If the results have opposite signs then it is a true maximum/minimum. You can also use these slopes to figure out if it is a maximum or a minimum: the left side slope will be positive for a maximum and negative for a minimum. However, you must exercise caution with this method, as, if you pick a point too far from the extremum, you could take it on the far side of another extremum and incorrectly classify the point.
The Extremum Test
A more rigorous method to classify a stationary point is called the extremum test, or 2nd Derivative Test. As we mentioned before, the sign of the first derivative must change for a stationary point to be a true extremum. Now, the second derivative of the function tells us the rate of change of the first derivative. It therefore follows that if the second derivative is positive at the stationary point, then the gradient is increasing. The fact that it is a stationary point in the first place means that this can only be a minimum. Conversely, if the second derivative is negative at that point, then it is a maximum.
Now, if the second derivative is zero, we have a problem. It could be a point of inflexion, or it could still be an extremum. Examples of each of these cases are below  all have a second derivative equal to zero at the stationary point in question:
 has a point of inflexion at
 has a minimum at
 has a maximum at
However, this is not an insoluble problem. What we must do is continue to differentiate until we get, at the (n+1)th derivative, a nonzero result at the stationary point:
If n is odd, then the stationary point is a true extremum. If the (n+1)th derivative is positive, it is a minimum; if the (n+1)th derivative is negative, it is a maximum. If n is even, then the stationary point is a point of inflexion.
As an example, let us consider the function
We now differentiate until we get a nonzero result at the stationary point at x=0 (assume we have already found this point as usual):
Therefore, (n+1) is 4, so n is 3. This is odd, and the fourth derivative is negative, so we have a maximum. Note that none of the methods given can tell you if this is a global extremum or just a local one. To do this, you would have to set the function equal to the height of the extremum and look for other roots.
Critical Points
Critical points are the points where a function's derivative is 0 or not defined. Suppose we are interested in finding the maximum or minimum on given closed interval of a function that is continuous on that interval. The extreme values of the function on that interval will be at one or more of the critical points and/or at one or both of the endpoints. We can prove this by contradiction. Suppose that the function has maximum at a point in the interval where the derivative of the function is defined and not . If the derivative is positive, then values slightly greater than will cause the function to increase. Since is not an endpoint, at least some of these values are in . But this contradicts the assumption that is the maximum of for in . Similarly, if the derivative is negative, then values slightly less than will cause the function to increase. Since is not an endpoint, at least some of these values are in . This contradicts the assumption that is the maximum of for in . A similar argument could be made for the minimum.
Example 1
Consider the function on the interval . The unrestricted function has no maximum or minimum. On the interval , however, it is obvious that the minimum will be , which occurs at and the maximum will be , which occurs at . Since there are no critical points ( exists and equals everywhere), the extreme values must occur at the endpoints.
Example 2
Find the maximum and minimum of the function on the interval .
 First start by finding the roots of the function derivative:
 Now evaluate the function at all critical points and endpoints to find the extreme values.
 From this we can see that the minimum on the interval is 24 when and the maximum on the interval is when
See "Optimization" for a common application of these principles.
<h1> 3.12 Newton's Method</h1>
Newton's Method (also called the NewtonRaphson method) is a recursive algorithm for approximating the root of a differentiable function. We know simple formulas for finding the roots of linear and quadratic equations, and there are also more complicated formulae for cubic and quartic equations. At one time it was hoped that there would be formulas found for equations of quintic and higherdegree, though it was later shown by Neils Henrik Abel that no such equations exist. The NewtonRaphson method is a method for approximating the roots of polynomial equations of any order. In fact the method works for any equation, polynomial or not, as long as the function is differentiable in a desired interval.
Let be a differentiable function. Select a point based on a first approximation to the root, arbitrarily close to the function's root. To approximate the root you then recursively calculate using: As you recursively calculate, the 's often become increasingly better approximations of the function's root. 
In order to explain Newton's method, imagine that is already very close to a zero of . We know that if we only look at points very close to then looks like its tangent line. If was already close to the place where was zero, and near we know that looks like its tangent line, then we hope the zero of the tangent line at is a better approximation then itself.
The equation for the tangent line to at is given by
Now we set and solve for .
This value of we feel should be a better guess for the value of where We choose to call this value of , and a little algebra we have
If our intuition was correct and is in fact a better approximation for the root of , then our logic should apply equally well at . We could look to the place where the tangent line at is zero. We call , following the algebra above we arrive at the formula
And we can continue in this way as long as we wish. At each step, if your current approximation is our new approximation will be
Examples
Find the root of the function .
As you can see is gradually approaching zero (which we know is the root of ). One can approach the function's root with arbitrary accuracy.
Answer: has a root at .
Notes
This method fails when . In that case, one should choose a new starting place. Occasionally it may happen that and have a common root. To detect whether this is true, we should first find the solutions of , and then check the value of at these places.
Newton's method also may not converge for every function, take as an example:
For this function choosing any then would cause successive approximations to alternate back and forth, so no amount of iteration would get us any closer to the root than our first guess.
Newton's method may also fail to converge on a root if the function has a local maximum or minimum that does not cross the xaxis. As an example, consider with initial guess . In this case, Newton's method will be fooled by the function, which dips toward the xaxis but never crosses it in the vicinity of the initial guess.
See also
 Wikipedia:Newton's method
 Wikibooks:Fractals/Mathematics/Newton_method
 Wikipedia:Abel–Ruffini theorem
<h1> 3.13 Related Rates</h1>
Introduction
One useful application of derivatives is as an aid in the calculation of related rates. What is a related rate? In each case in the following examples the related rate we are calculating is a derivative with respect to some value. We compute this derivative from a rate at which some other known quantity is changing. Given the rate at which something is changing, we are asked to find the rate at which a value related to the rate we are given is changing.
Process for solving related rates problems:
 Write out any relevant formulas and information.
 Take the derivative of the primary equation with respect to time.
 Solve for the desired variable.
 Plugin known information and simplify.
Notation
Newton's dot notation is used to show the derivative of a variable with respect to time. That is, if is a quantity that depends on time, then , where represents the time. This notation is a useful abbreviation in situations where time derivatives are often used, as is the case with related rates.
Examples
Example 1:
 Write out any relevant formulas or pieces of information.
 Take the derivative of the equation above with respect to time. Remember to use the Chain Rule and the Product Rule.
Example 2:
 Write out any relevant formulas and pieces of information.
 Take the derivative of both sides of the volume equation with respect to time.

= =
 Solve for .
 Plugin known information.
Example 3:
Note: Because the vertical distance is downward in nature, the rate of change of y is negative. Similarly, the horizontal distance is decreasing, therefore it is negative (it is getting closer and closer).
The easiest way to describe the horizontal and vertical relationships of the plane's motion is the Pythagorean Theorem.
 Write out any relevant formulas and pieces of information.
 (where s is the distance between the plane and the house)
 Take the derivative of both sides of the distance formula with respect to time.
 Solve for .

=
 Plugin known information

= = = ft/s
Example 4:
 Write down any relevant formulas and information.
Substitute into the volume equation.

= = =
 Take the derivative of the volume equation with respect to time.
 Solve for .
 Plugin known information and simplify.

= = ft/min
Example 5:
 Write out any relevant formulas and information.
Use the Pythagorean Theorem to describe the motion of the ladder.
 (where l is the length of the ladder)
 Take the derivative of the equation with respect to time.
 ( is constant so .)
 Solve for .
 Plugin known information and simplify.

= = ft/sec
Exercises
<h1> 3.14 Optimization</h1>
Introduction
Optimization is one of the uses of calculus in the real world. Let us assume we are a pizza parlor and wish to maximize profit. Perhaps we have a flat piece of cardboard and we need to make a box with the greatest volume. How does one go about this process?
This requires the use of maximums and minimums. We know that we find maximums and minimums via derivatives. Therefore, one can conclude that calculus will be a useful tool for maximizing or minimizing (collectively known as "optimizing") a situation.
How to Solve
These general steps should be taken in order to complete an optimization problem.
 Write out formulas and other pieces of information about the problem.
 The problems should have a variable you control and a variable that you want to maximize/minimize.
 The formulas you find may contain extra variables. Depending on how the question works out, they may be substituted out or can be ignored (which will be explained later).
 Combine the formulas together so that the variable you want to maximize/minimize is on one side of the equation and everything else is on the other side.
 Differentiate the formula. If your equation has multiple variables, pick any variable to differentiate as long as it is not the one you control (i.e. pick the variable that you could not get rid of from the formula).
 Note that during differentiation, if you come across a variable that you have not picked, imagine it as a number and apply the necessary differentiation rule. Do not treat it as a variable in this case.
 Set the differentiated formula to equal 0 and solve for the variable you control.
 The value you get here is your answer. If you instead have another formula, that means that your answer depends on those other variables, which would usually be what the question asked for if you have such a situation that you have another variable to juggle to begin with.
The reason why this algorithm works comes from a few mathematical theorems which you will probably not need to know when completing these problems. Usually the problems given will be mathematically simple (in other words, there are not a lot of cases to test). However, if you wish to know, they work like this:
 A derivative of 0 is either a global or local maximum or minimum. Usually the question will tend towards answering that question without much difficulty (like always positive numbers, for example)
Examples
Volume Example
A box manufacturer desires to create a box with a surface area of 100 inches squared. What is the maximum size volume that can be formed by bending this material into a box? The box is to be closed. The box is to have a square base, square top, and rectangular sides.
 Write out known formulas and information
 Eliminate the variable h in the volume equation
 Find the derivative of the volume equation in order to maximize the volume
 Set and solve for
 Plugin the x value into the volume equation and simplify
Answer:
Volume Example II
It is desired to make an opentop box of greatest possible volume from a square piece of tin whose side is , by cutting equal squares out of the corners and then folding up the tin to form the sides. What should be the length of a side of the squares cut out?
If we call the side length of the cut out squares , then each side of the base of the folded box is , and the height is . Therefore, the volume function is .
We must optimize the volume by taking the derivative of the volume function and setting it equal to 0. Since it does not change, is treated as a constant, not a variable.
We can now use the quadratic formula to solve for :
We reject , since it is a minimum (it results in the base length being 0, making the volume 0). Therefore, the answer is .
Sales Example
A small retailer can sell n units of a product for a revenue of and at a cost of , with all amounts in thousands. How many units does it sell to maximize its profit?
The retailer's profit is defined by the equation , which is the revenue generated less the cost. The question asks for the maximum amount of profit which is the maximum of the above equation. As previously discussed, the maxima and minima of a graph are found when the slope of said graph is equal to zero. To find the slope one finds the derivative of . By using the subtraction rule :
Therefore, when the profit will be maximized or minimized. Use the quadratic formula to find the roots, giving {3.798,0.869}. To find which of these is the maximum and minimum the function can be tested:
Because we only consider the functions for all (i.e., you can't have units), the only points that can be minima or maxima are those two listed above. To show that 3.798 is in fact a maximum (and that the function doesn't remain constant past this point) check if the sign of changes at this point. It does, and for n greater than 3.798 the value will remain decreasing. Finally, this shows that for this retailer selling 3.798 units would return a profit of $8,588.02.
<h1> 3.15 Euler's Method</h1>
Euler's Method is a method for estimating the value of a function based upon the values of that function's first derivative.
The general algorithm for finding a value of is:
where f is y'(x). In other words, the new value, , is the sum of the old value and the step size times the change, .
You can think of the algorithm as a person traveling with a map: Now I am standing here and based on these surroundings I go that way 1 km. Then, I check the map again and determine my direction again and go 1 km that way. I repeat this until I have finished my trip.
The Euler method is mostly used to solve differential equations of the form
Examples
A simple example is to solve the equation:
This yields and hence, the updating rule is:
Step size = 0.1 is used here.
The easiest way to keep track of the successive values generated by the algorithm is to draw a table with columns for .
The above equation can be e.g. a population model, where y is the population size and x is time.
<h1> 3.16 Applications of Derivatives Cumulative Exercises</h1>
Relative Extrema
Find the relative maximum(s) and minimum(s), if any, of the following functions.
Range of Function
Absolute Extrema
Determine the absolute maximum and minimum of the following functions on the given domain
Determine Intervals of Change
Find the intervals where the following functions are increasing or decreasing
Determine Intervals of Concavity
Find the intervals where the following functions are concave up or concave down
Word Problems
Graphing Functions
For each of the following, graph a function that abides by the provided characteristics
Integration
Basics of Integration
<h1> 4.1 Definite Integral</h1>
Suppose we are given a function and would like to determine the area underneath its graph over an interval. We could guess, but how could we figure out the exact area? Below, using a few clever ideas, we actually define such an area and show that by using what is called the definite integral we can indeed determine the exact area underneath a curve.
Definition of the Definite Integral
The rough idea of defining the area under the graph of is to approximate this area with a finite number of rectangles. Since we can easily work out the area of the rectangles, we get an estimate of the area under the graph. If we use a larger number of smallersized rectangles we expect greater accuracy with respect to the area under the curve and hence a better approximation. Somehow, it seems that we could use our old friend from differentiation, the limit, and "approach" an infinite number of rectangles to get the exact area. Let's look at such an idea more closely.
Suppose we have a function that is positive on the interval and we want to find the area under between and . Let's pick an integer and divide the interval into subintervals of equal width (see Figure 1). As the interval has width , each subinterval has width We denote the endpoints of the subintervals by . This gives us
Now for each pick a sample point in the interval and consider the rectangle of height and width (see Figure 2). The area of this rectangle is . By adding up the area of all the rectangles for we get that the area is approximated by
A more convenient way to write this is with summation notation:
For each number we get a different approximation. As gets larger the width of the rectangles gets smaller which yields a better approximation (see Figure 3). In the limit of as tends to infinity we get the area .
Definition of the Definite Integral
Suppose is a continuous function on and . Then the definite integral of between and is
It is a fact that if is continuous on then this limit always exists and does not depend on the choice of the points . For instance they may be evenly spaced, or distributed ambiguously throughout the interval. The proof of this is technical and is beyond the scope of this section.
Notation
One important feature of this definition is that we also allow functions which take negative values. If for all then so . So the definite integral of will be strictly negative. More generally if takes on both positive an negative values then will be the area under the positive part of the graph of minus the area above the graph of the negative part of the graph (see Figure 4). For this reason we say that is the signed area under the graph.
Independence of Variable
It is important to notice that the variable did not play an important role in the definition of the integral. In fact we can replace it with any other letter, so the following are all equal:
Each of these is the signed area under the graph of between and . Such a variable is often referred to as a dummy variable or a bound variable.
Left and Right Handed Riemann Sums
The following methods are sometimes referred to as LRAM and RRAM, RAM standing for "Rectangular Approximation Method."
We could have decided to choose all our sample points to be on the right hand side of the interval (see Figure 5). Then for all and the approximation that we called for the area becomes
This is called the righthanded Riemann sum, and the integral is the limit
Alternatively we could have taken each sample point on the left hand side of the interval. In this case (see Figure 6) and the approximation becomes
Then the integral of is
The key point is that, as long as is continuous, these two definitions give the same answer for the integral.
Examples
Example 1
In this example we will calculate the area under the curve given by the graph of for between 0 and 1. First we fix an integer and divide the interval into subintervals of equal width. So each subinterval has width
To calculate the integral we will use the righthanded Riemann sum. (We could have used the lefthanded sum instead, and this would give the same answer in the end). For the righthanded sum the sample points are
Notice that . Putting this into the formula for the approximation,
Now we use the formula
to get
To calculate the integral of between and we take the limit as tends to infinity,
Example 2
Next we show how to find the integral of the function between and . This time the interval has width so
Once again we will use the righthanded Riemann sum. So the sample points we choose are
Thus
We have to calculate each piece on the right hand side of this equation. For the first two,
For the third sum we have to use a formula
to get
Putting this together
Taking the limit as tend to infinity gives
Exercises
Basic Properties of the Integral
From the definition of the integral we can deduce some basic properties. For all the following rules, suppose that f and g are continuous on [a,b].
The Constant Rule
Constant Rule
When f is positive, the height of the function cf at a point x is c times the height of the function f. So the area under cf between a and b is c times the area under f. We can also give a proof using the definition of the integral, using the constant rule for limits,
Example
We saw in the previous section that
 .
Using the constant rule we can use this to calculate that
Example
We saw in the previous section that
We can use this and the constant rule to calculate that
There is a special case of this rule used for integrating constants:
Integrating Constants
When and this integral is the area of a rectangle of height c and width ba which equals c(ba).
Example
The addition and subtraction rule
Addition and Subtraction Rules of Integration
As with the constant rule, the addition rule follows from the addition rule for limits:

= = =
The subtraction rule can be proved in a similar way.
Example
From above and so
Example
Exercise
The Comparison Rule
Comparison Rule
 Suppose for all x in [a,b]. Then
 Suppose for all x in [a,b]. Then
 Suppose for all x in [a,b]. Then
If then each of the rectangles in the Riemann sum to calculate the integral of f will be above the y axis, so the area will be nonnegative. If then and by the first property we get the second property. Finally if then the area under the graph of f will be greater than the area of rectangle with height m and less than the area of the rectangle with height M (see Figure 7). So
Linearity with respect to endpoints
Additivity with respect to endpoints Suppose . Then
Again suppose that is positive. Then this property should be interpreted as saying that the area under the graph of between and is the area between and plus the area between and (see Figure 8).
Extension of Additivity with respect to limits of integration
When we have that so
Also in defining the integral we assumed that . But the definition makes sense even when , in which case has changed sign. This gives
With these definitions,
Exercise
Even and odd functions
Recall that a function is called odd if it satisfies and is called even if
Suppose is a continuous odd function then for any ,
If is a continuous even function then for any ,
Suppose is an odd function and consider first just the integral from to . We make the substitution so . Notice that if then and if then . Hence Now as is odd, so the integral becomes Now we can replace the dummy variable with any other variable. So we can replace it with the letter to give
Now we split the integral into two pieces
The proof of the formula for even functions is similar.
<h1> 4.2 Fundamental Theorem of Calculus</h1>
The fundamental theorem of calculus is a critical portion of calculus because it links the concept of a derivative to that of an integral. As a result, we can use our knowledge of derivatives to find the area under the curve, which is often quicker and simpler than using the definition of the integral.
Mean Value Theorem for Integration
We will need the following theorem in the discussion of the Fundamental Theorem of Calculus.
Mean Value Theorem for Integration
Suppose is continuous on . Then for some in .Proof of the Mean Value Theorem for Integration
satisfies the requirements of the Extreme Value Theorem, so it has a minimum and a maximum in . Since
and since
for all in ,
we have
Since is continuous, by the Intermediate Value Theorem there is some with in such that
Fundamental Theorem of Calculus
Statement of the Fundamental Theorem
Suppose that f is continuous on [a,b]. We can define a function F by
Fundamental Theorem of Calculus Part I Suppose f is continuous on [a,b] and F is defined by
Then F is differentiable on (a,b) and for all ,
When we have such functions and where for every in some interval we say that is the antiderivative of on .
Fundamental Theorem of Calculus Part II Suppose that f is continuous on [a,b] and that F is any antiderivative of f. Then
Note: a minority of mathematicians refer to part one as two and part two as one. All mathematicians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.
Proofs
Proof of Fundamental Theorem of Calculus Part I
Suppose x is in (a,b). Pick so that is also in (a, b). Then
and
 .
Subtracting the two equations gives
Now
so rearranging this we have
According to the Mean Value Theorem for Integration, there exists a c in [x, x + Δx] such that
 .
Notice that c depends on . Anyway what we have shown is that
 ,
and dividing both sides by Δx gives
 .
Take the limit as we get the definition of the derivative of F at x so we have
 .
To find the other limit, we will use the squeeze theorem. The number c is in the interval [x, x + Δx], so x≤ c ≤ x + Δx. Also, and . Therefore, according to the squeeze theorem,
 .
As f is continuous we have
which completes the proof.
Proof of Fundamental Theorem of Calculus Part II
Define Then by the Fundamental Theorem of Calculus part I we know that is differentiable on and for all
So is an antiderivative of . Since we were assuming that was also an antiderivative for all ,
 .
Let . The Mean Value Theorem applied to on with says that
for some in . But since for all in , must equal for all in , i.e. g(x) is constant on .
This implies there is a constant such that for all ,
 ,
and as is continuous we see this holds when and as well. And putting gives
Notation for Evaluating Definite Integrals
The second part of the Fundamental Theorem of Calculus gives us a way to calculate definite integrals. Just find an antiderivative of the integrand, and subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound. That is
where . As a convenience, we use the notation
to represent .
Integration of Polynomials
Using the power rule for differentiation we can find a formula for the integral of a power using the Fundamental Theorem of Calculus. Let . We want to find an antiderivative for . Since the differentiation rule for powers lowers the power by 1 we have that
As long as we can divide by to get
So the function is an antiderivative of . If is not in then is continuous on and, by applying the Fundamental Theorem of Calculus, we can calculate the integral of to get the following rule.
Power Rule of Integration I
Notice that we allow all values of , even negative or fractional. If then this works even if includes .
Power Rule of Integration II
Examples
 To find we raise the power by 1 and have to divide by 4. So
 The power rule also works for negative powers. For instance
 We can also use the power rule for fractional powers. For instance
 Using linearity the power rule can also be thought of as applying to constants. For example,
 .
 Using the linearity rule we can now integrate any polynomial. For example
Exercises
<h1> 4.3 Indefinite Integral</h1>
Definition
Now recall that F is said to be an antiderivative of f if . However, F is not the only antiderivative. We can add any constant to F without changing the derivative. With this, we define the indefinite integral as follows:
The function , the function being integrated, is known as the integrand. Note that the indefinite integral yields a family of functions.
Example
Since the derivative of is , the general antiderivative of is plus a constant. Thus,
Example: Finding antiderivatives
Let's take a look at . How would we go about finding the integral of this function? Recall the rule from differentiation that
In our circumstance, we have:
This is a start! We now know that the function we seek will have a power of 3 in it. How would we get the constant of 6? Well,
Thus, we say that is an antiderivative of .
Exercises
Indefinite integral identities
Basic Properties of Indefinite Integrals
Constant Rule for indefinite integrals
Sum/Difference Rule for indefinite integrals
Indefinite integrals of Polynomials
Say we are given a function of the form, , and would like to determine the antiderivative of f. Considering that
we have the following rule for indefinite integrals:
Power rule for indefinite integrals
for allIntegral of the Inverse function
To integrate , we should first remember
Therefore, since is the derivative of we can conclude that
Note that the polynomial integration rule does not apply when the exponent is 1. This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.
Integral of the Exponential function
Since
we see that is its own antiderivative. This allows us to find the integral of an exponential function:
Integral of Sine and Cosine
Recall that
So sin x is an antiderivative of cos x and cos x is an antiderivative of sin x. Hence we get the following rules for integrating sin x and cos x
We will find how to integrate more complicated trigonometric functions in the chapter on integration techniques.
Example
Suppose we want to integrate the function . An application of the sum rule from above allows us to use the power rule and our rule for integrating as follows,
Exercises
The Substitution Rule
The substitution rule is a valuable asset in the toolbox of any integration greasemonkey. It is essentially the chain rule (a differentiation technique you should be familiar with) in reverse. First, let's take a look at an example:
Preliminary Example
Suppose we want to find . That is, we want to find a function such that its derivative equals . Stated yet another way, we want to find an antiderivative of . Since differentiates to , as a first guess we might try the function . But by the Chain Rule,
Which is almost what we want apart from the fact that there is an extra factor of 2 in front. But this is easily dealt with because we can divide by a constant (in this case 2). So,
Thus, we have discovered a function, , whose derivative is . That is, F is an antiderivative of . This gives us
Generalization
In fact, this technique will work for more general integrands. Suppose u is a differentiable function. Then to evaluate we just have to notice that by the Chain Rule
As long as is continuous we have that
Now the right hand side of this equation is just the integral of but with respect to u. If we write u instead of u(x) this becomes
So, for instance, if we have worked out that
General Substitution Rule
Now there was nothing special about using the cosine function in the discussion above, and it could be replaced by any other function. Doing this gives us the substitution rule for indefinite integrals:
Substitution rule for indefinite integrals
Assume u is differentiable with continuous derivative and that f is continuous on the range of u. Then
Notice that it looks like you can "cancel" in the expression to leave just a . This does not really make any sense because is not a fraction. But it's a good way to remember the substitution rule.
Examples
The following example shows how powerful a technique substitution can be. At first glance the following integral seems intractable, but after a little simplification, it's possible to tackle using substitution.
Example
We will show that
First, we rewrite the integral:

 .
Now we preform the following substitution:
Which yields: