Supplementary mathematics/Differential calculus

Differential calculus is a branch of differential and integral calculus and is a part of mathematical analysis. This subject was invented by Isaac Newton, an English scientist, and JW Leibniz, a German scientist. Therefore, it involves calculating derivatives and using them to solve problems related to non-constant rates of change.It has other applications such as the maximum and minimum values of functions and the performance of the function in the form of the limit of the derivative.

The primary and basic subjects of studying in differential calculus are the derivative of functions and their function regarding the continuity of the function and the concepts related to differential calculus and their applications. The derivative of a function is selected at the input of the function and then its output is considered the result of the derivative of the same function,Of course, it can be said that the continuity and function of the rate of change of a function can be described as the input value of the same function.

The function process definition is like this:

The process of finding the derivative for functions is called differentiation. Of course, if the function is complex, the solution of the process of finding its derivative is expanded, and the concepts of the process are created in showing the function as a complex pattern in the graphs, and its generalization should be as a minimum or be the maximum, of course, it depends on the limit of the function in the derivative.

The derivative in terms of geometry is that at a point, the slope of the tangent line is on the curved or broken graph of the function at the same point. Of course, the determination of the sign of the tangent line depends on the lower and upper slope of the function. If the slope is low, the tangent line is descending. and if the slope is high, the tangent line of the derivative is ascending.Of course, the topic of derivative tangent is provided that the derivative exists and is defined at that point.

For a real-valued function of a real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Calculus and calculus are connected by the Fundamental Theorem of Calculus, which states that differentiation is the inverse process of integration.

Differentiation is used in almost all quantitative disciplines. In physics, the derivative of the displacement of a moving object with respect to time is the velocity of the object and the derivative of the velocity with respect to time is the acceleration. The derivative of the momentum of an object with respect to time is equal to the force applied to the object. Rearranging this derivative expression leads to the famous equation F = ma associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transfer materials and design plants.

Derivatives are often used to find the maximum and minimum of a function. Derivative equations are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many areas of mathematics such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.

Introduction

Introduction based on an example

If a car is driving on a road, a table can be created on the basis of this fact, in which the distance that has been covered since the start of the recording is entered at each point in time. In practice, it makes sense not to keep such a table too dense, i.e. for example, to make a new entry only every 3 seconds in a period of 1 minute, which would only require 20 measurements. However, such a table can theoretically be made as tight as you like if every point in time is to be taken into account. The previously discrete data, i.e. data with a distance, merge into a continuum. The present is then used as a point, i.e. interpreted as an infinitely short period of time. At the same time, however, the car has covered a theoretically known distance at any point in time, and if it does not brake to a standstill or even reverse, the distance will increase continuously, i.e. it will never be the same as at any other point in time.

• Example representation of a table, every 3 seconds a new measurement is entered. Under such conditions, only average speeds in the periods 0 to 3, 3 to 6, etc. seconds can be calculated. Since the distance traveled is always increasing, the car only seems to go forward.

  
  Example representation of a table, every 3 seconds a new measurement is entered. Under such conditions, only average speeds in the periods 0 to 3, 3 to 6, etc. seconds can be calculated. Since the distance traveled is always increasing, the car only seems to go forward.

• Transition to an arbitrarily close-meshed table, which takes the form of a curve after all points have been entered. Each point in time between 0 and 60 seconds is assigned a point on the curve. Regions within which the curve is steeper upwards correspond to periods of time in which a greater distance is traveled per period of time. In regions where the distance is almost constant, for example in the 15-20 second range, the car drives slowly and the curve is flat.

  
  Transition to an arbitrarily close-meshed table, which takes the form of a curve after all points have been entered. Each point in time between 0 and 60 seconds is assigned a point on the curve. Regions within which the curve is steeper upwards correspond to periods of time in which a greater distance is traveled per period of time. In regions where the distance is almost constant, for example in the 15-20 second range, the car drives slowly and the curve is flat.

The motivation behind the notion of derivative of a path-time curve or function is that it can now be specified how fast the car is moving at a momentary point in time. The appropriate speed-time curve should be derived from a distance-time curve. The background is that speed is a measure of how much the distance covered changes over time. At high speed there is a sharp rise in the curve, while at low speed there is little change. Since each measuring point was also assigned a route, such an analysis should be possible in principle, because with the knowledge of the distance covered $\Delta s$ within a time interval $\Delta t</math>appliestothespeed:<math>v={\frac {\Delta s}{\Delta t}}.$

So if $t_{0}$ and $t_{1}$ are two different points in time, then this is the "speed" of the car in the time interval between them

$v={\frac {s(t_{1})-s(t_{0})}{t_{1}-t_{0}}}.$

The differences in the numerator and denominator must be formed, since one is only interested in the distance $s(t_{1})-s(t_{0})$ covered within a certain time interval $t_{1}-t_{0}$. Nevertheless, this approach does not provide a complete picture, since initially only speeds for time intervals with a different start and end point were measured. A instantaneous speed, comparable to a speed camera, on the other hand, would refer to an infinitely short time interval. Accordingly, the above term "speed" should be specified by "average speed". Even when working with real time intervals, i.e. discrete data, the model is simplified to the extent that no sudden change in location or speed is possible for a car within the intervals considered. (Emergency braking also takes time, and indeed longer than the time in which the tires squeak.) This also justifies the tacitly continuously entered curve in the drawing without any jumps or kinks. At the timepoint 25 seconds, the car is currently moving at about 7.6 meters per second, the equivalent of 27 km/h. This value corresponds to the gradient of the tangent of the path-time curve at the corresponding point. Further more detailed explanations of this geometric interpretation are given below.

On the other hand, if you want to switch to a "perfectly fitting" speed-time curve, the term "average speed in a time interval" must be replaced by "speed at a point in time". To do this, a point in time $t_{0}$ must first be selected. The idea is well, run "extended time intervals" in a limit process against an infinitely short time interval and study what happens at the average speeds involved. Although the denominator $t_{1}-t_{0}$ tends towards 0, this is not a problem, since the car can move less and less far in the continuous progression in shorter periods of time, which means that The numerator and denominator decrease at the same time, and an indefinite term "${\tfrac {0}{0}}$" is created in the limit process. Under certain circumstances, this can make sense as a limit value, for example pressing

${\tfrac {5\ \mathrm {meter} }{\mathrm {second} }}\ {\text{ and }}\ {\tfrac {5\ \mathrm {millimeter} }{\mathrm {millisecond} }}\ {\text{ and }}\ {\tfrac {5\ \mathrm {nanometer} }{\mathrm {nanosecond} }}\ {\text{ etc.}}$

exactly the same speeds. Now there are two possibilities when studying velocities. Either, they show no tendency to approach a certain finite value in the considered limit value process. In this case, the movement of the car cannot be assigned a speed valid at the time $t_{0}$, i.e. the term "${\tfrac {0}{0}}$" has no clear meaning here. If, on the other hand, there is increasing stabilization towards a fixed value, then the limit value exists.

${\frac {\mathrm {d} s}{\mathrm {d} t}}(t_{0}):=\lim _{t_{1}\to t_{0}}{\frac {s(t_{1})-s(t_{0})}{t_{1}-t_{0}}}=\lim _{\Delta t\to 0}{\frac {s(t_{0}+\Delta t)-s(t_{0})}{\Delta t}}$

and expresses the velocity existing exactly at $t_{0}$. In this case, the indefinite term "${\tfrac {0}{0}}$" takes on a unique value. The resulting instantaneous speed is also referred to as the derivative of $s$ at the point $t_{0}$; for these the symbol $s'(t_{0})$ is often used. The limit value defines the instantaneous speed at any point in time as

$v={\frac {\mathrm {d} s}{\mathrm {d} t}}.$

Principle of differential calculus

The example of the last section is particularly simple if the increase in the distance traveled over time is uniform, i.e. linear. Then there is specifically a proportionality between the change of distance and the change of time. The relative change of the distance, ie its increase in relation to the increase in time, is always the same with this movement. The average speed at any point in time is also the instantaneous speed. For example, the car travels the same distance between 0 and 1 second as between 9 and 10 seconds and ten times the distance between 0 and 10 seconds. The constant speed $v$ applies as a proportionality factor over the entire path, whereby it is $v=2\,\mathrm {m/s}$ in the adjacent figure. The distance covered between $t$ and $t+\Delta t$ that are arbitrarily far apart is

$\Delta s=s(t+\Delta t)-s(t)=v\cdot (t+\Delta t)-v\cdot t=v\cdot \Delta t$.

In general, the car moves forward in the time span $\Delta t$ by the distance $\Delta s=v\,\Delta t$. Especially with $\Delta t=5\,\mathrm {s}$ there is a stretch $\Delta s=v\,\Delta t=2\,\mathrm {{\tfrac {m}{s}}\cdot 5\ ,s=10\,m}$.

If the starting value at $t=0$ is not $s(0)=0$ but $s(0)=c\neq 0$, this changes nothing , since in the relation $s=v\,t+c$ the constant $c$ is derived from $\Delta s$ is always subtracted out. This is also well known: The starting position of the car is irrelevant for its speed.

Instead of the Variables $t$ and $s$, the variables $x$ and $y$ are generally used considered, it can be stated that:

• Linear Functions: With Linearity the considered function has the form $y=f(x)=mx+c$. (A line through the origin is not necessary for a linear function!) The relative change applies here as a derivation, in other words the difference quotient ${\tfrac {\Delta y}{\Delta x}}$. It has the same value at every point $m$. The derivation can be read directly from the expression $mx+c$. In particular, every constant function $f(x)=c$ has the derivative ${\tfrac {\Delta y}{\Delta x}}=0$, since with a Changing the input value does not change the output value.

It becomes more difficult when a movement is non-uniform. Then the diagram of the time-distance function is not straight. The term derivation must be expanded for such courses. Because there is no proportionality factor that expresses the local relative change everywhere. The only possible strategy has been found to be to obtain a linear approximation for the non-linear function, at least at one point of interest. (In the next figure, this is the point $x=1$.) This reduces the problem to a function that is linear at least at this point. The method of linearization is the basis for the actual calculus of differential calculus. It is of very great importance in Analysis, since it helps to reduce complicated processes locally to processes that are easier to understand, namely linear processes.

$x$	0.5	0.9	0.99	0.999	1	1.001	1.01	1.1	1.5	2
$f=x^{2}$	0.25	0.81	0.9801	0.998001	1	1.002001	1.0201	1.21	2.25	4
$g=2x-1$	0	0.8	0.98	0.998	1	1.002	1.02	1.2	2	3
$g(x)-f(x)$	−0.25	−0.01	−0.0001	−0.000001	0	−0.000001	−0.0001	−0.01	−0.25	−1
${\Big |}{\tfrac {g(x)-f(x)}{x-1}}{\Big |}$	50%	10%	1%	|	0.1%	1%	10%	50%	100%

The strategy is to be explained using the non-linear function $f(x)=x^{2}$ as an example. The table shows values for this function and for its approximation function at $x=1$, which is $g(x)=2x-1$. The table below contains the deviation hung the approximation from the original function. (The values are negative because in this case the straight line is always under the curve - except at the point of contact.) The last line contains the amount of the relative deviation, which is the deviation based on the Distance of location $x$ from the touch point at $x=1$. This cannot be calculated at the point of contact. But the surrounding values show the relative deviation approaching a limit, here zero. This zero means: Even if $x$ moves a little (infinitesimal) away from the point of contact, there is still no difference between $g(x)$ and $f(x)$.

The linear function $g(x)$ closely mimics the behavior of $f(x)$ near $x=1$ (better than any other linear function Function). The relative change ${\tfrac {\Delta g}{\Delta x}}$ has the value $m=2$ everywhere. The relative change ${\tfrac {\Delta f}{\Delta x}}$, which is not so easy to determine, agrees with the value $m=2$ at the point of contact.

So it can be stated:

• Non-linear functions: If the relative change in a non-linear function is to be determined at a certain point, it is (if possible) linearly approximated there. The gradient of the linear approximation function is the gradient of the non-linear function under consideration at this point, and the same view applies as for derivatives of linear functions. The only thing to note is that the relative change of a non-linear function changes from point to point.

While in the example above (vehicle motion) for the average velocity the time span $\Delta t$ can be chosen reasonably arbitrarily, the instantaneous velocity, if variable, is $onlyfor''small''ones\Delta t$ can be specified. How small $\Delta t$ has to be chosen depends on the quality requirements of the approximation. In mathematical perfection it becomes infinitesimal. In this case, for the relative change (as already stated above), the differential quotient ${\tfrac {\mathrm {d} y}{\mathrm {d} x}}$ (in simplified notation $y'$ or $f'$).

Obtaining the linear approximation of a non-linear function at a certain point is the central task of the calculus of differential calculus. With a mathematically definable function (in the example it was $f(x)=x^{2}$) the derivative should be calculate. In the ideal case, this calculation is even so general that it can be applied to all points of the domain of definition. In the case of $f(x)=x^{2}$ it can be shown that at any point $x$ the best linear approximation is the slope $m=2x$ must own. With the additional information that the linear function must match the curve at the point $(x_{0},f(x_{0}))$, the complete function equation of the linear approximation function can then be set up.

The approach to determining the differential quotient lies in the calculation of the limit value (as with the instantaneous speed above):

$\lim _{\Delta x\to 0}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}=f'(x_{0})\quad$ or in different notation $\quad \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=f'(x).$

For some elementary functions such as power function, exponential function, logarithmic function or sine function, the limit value process has been carried out in each case. This results in a derivative function. Building on this, there are derivation rules for the elementary and also for other functions such as sumn, products or concatenations elementary functions have been set up.

This means that the limit crossings are not carried out anew in each application, but derivation rules are used for the calculation practice. The "art" of differential calculus consists "only" in structuring more complicated functions and applying the appropriate derivation rule to the structural elements. An example follows further down.