Real Analysis/Differentiation

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Real Analysis
Differentiation

In this chapter, we will introduce the concept of differentiation. Differentiation is a staple tool in calculus, which should be a fact somewhat familiar to you from studying earlier mathematics. However, the reasons as to why this is true have not always been so clearly proven. This chapter will thus also prove a simple consequence of differentiation you will be most familiar with - that is, we will focus on proving each differentiation "operations" that provides us a simple way to find the derivative of common functions.

Definition[edit]

Let us define the derivative of a function

Given a function

Let

We say that ƒ(x) is differentiable at x=a if and only if

exists.

The derivative of ƒ at a and is denoted by

A function is said to be differentiable on a set A if the derivative exists for each a in A. A function is differentiable if it is differentiable on its entire domain.

Definition-Derived Theorems (Differentiability Implies Continuity)[edit]

Given our definition of a derivative, it should be noted that it utilizes limits and functions. This theorem relates derivation with continuity, which is useful for justifying many of the latter theorems that will be discussed in this chapter. The proof for this theorem is simple; it requires a valid limit converging to zero to mimic the continuity definition.

Theorem

Given a function ƒ which is differentiable at a, it is also continuous at a

The proof for this is as follows

Proof that Differentiable Functions at a are Continuous at a
Differentiable definition:
Suppose we multiply both sides by 0
Algebraic manipulations
The continuity definition

However, the converse is not true in this case. Analyzing the proof, it is apparent that a continuous function at a does not necessarily mean that it is differentiable at a simply because it would involve removing the multiplication by 0, which is impossible given our algebraic axioms.

Properties of Differentiation[edit]

From this definition, we will create new properties of derivation. People familiar with Calculus should note that we are proving that the derivation of certain functions and operations are valid. These first theorems follow immediately from the definition.

Basic Properties[edit]

Below are the list of properties which are mentioned only for completeness, and a demonstration of how the derivation formula works.

List of Basic Derivative
Constant Function Given
Identity Function Given

Constant Function

Suppose a constant function ƒ such that . This function will always have a derivative of 0 for any real number.

Given the derivation definition, we have

Applying the definition of the function, we can substitute the function for c, as such

Identity Function

Suppose an identity function ƒ such that . This function will always have a derivative of 1 for any real number.

Given the derivation definition, we have

Applying the definition of the function, we can substitute the function for its input, as such

Algebraic Properties[edit]

Suppose two functions f and g that are differentiable at a, these following properties apply

List of Derivatives for Algebraic Operations
Addition
Subtraction
Product
Multiple of a Function
Reciprocal
Division

We will individually prove each one below

Addition

This proof essentially creates the definition of differentiation from the two functions that make up the overall function.

Subtraction

We will not write out a rigorous proof for subtraction, given that it can be done mentally by imagining a negated function or retracing the addition proof with subtraction instead.

Product

This proof works similarly to the previous proof, except that this proof requires the addition of extra terms which zero out when added together. This is a normal algebraic trick in order to derive theorems, which will be further used in the latter theorems in this chapter.

Product of a Constant

For this proof, we will present it using two different methods.

The first method requires only the limit theorem that a constant multiple is equivalent to the limit being multiplied by the constant.

The second proof requires applying the product rule and constant function for differentiation.

Reciprocal

Like the other proofs before, this one will also invoke the definition at a certain point to simplify the statement into a concise, memorizable format.

Division

This proof borrows the reciprocal proof and the multiplication proof to form an easy to follow rationale.

Chain Rule (Function Composition Theorem)[edit]

Given two functions f and g such that f is differentiable at and g is differentiable at a, then

Proof Part 1

Unlike the previous properties, the chain rule will quickly become problematic and will definitely require an external theorem outside of algebraic manipulations to solve. To illustrate why a new theorem is required, we will begin to prove the Chain Rule though algebraic manipulations, point out the road block, then create a lemma to guide us around the issue, and thus figure out a proof. We begin with the following statements:


The problem is that may be zero at points arbitrarily close to x, and therefore would not be continuous at these points. Thus we apply a clever lemma as follows:

Caratheodory's Lemma[edit]

Let

We say that is differentiable at if and only if there exists a continuous function that satisfies

Proof

()Let be differentiable at and define function such that

for and

It is easy to see that is continuous and that it satisfies the required condition.

() Let be a continuous function satisfying

For all , we have that

As is continuous, , that is,

which implies that is differentiable at

Proof Part 2

Let be differentiable at

Let be differentiable at

Let the domain of f be a subset of the image of g.

Then,

  1. is differentiable at

Caratheodory's Lemma implies that there exist continuous functions such that and

Now, consider the function . Obviously, is continuous.

Also, it satisfies . Hence, by Caratheodory's Lemma, is differentiable at and that

Exercises[edit]

Here are some exercises to expand and train your understanding of the material.

  1. Find the derivatives of the following functions:
    1. A function of the form ƒ(x) = xn
    2. Polynomial
    3. Trigonometric
    4. Exponential
    5. Logarithmic
  2. In this chapter you have learned that being able to take the derivative implies that the function is continuous at that point. Given this, please read Higher Order Derivatives before solving these problems
    1. Prove whether that the second derivative at a is also continuous at a
    2. Prove whether that the nth derivative at a is also continuous at a
  3. Some of the most popular counter examples to illustrate properties of continuity and differentiability are functions involving
    1. Prove that is not continuous at
    2. Prove that the function is continuous but not differentiable at
    3. Prove that is differentiable at