# Real Analysis/Applications of Derivatives

←Differentiation | Real AnalysisApplications of Derivatives |
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## Definition[edit]

## Higher Order Derivatives[edit]

Let be differentiable

Let be differentiable for all . Then, the derivative of is called the **second derivative** of and is written as

Similarly, we can define the **n ^{th}-derivative** of , written as

## Theorem(Rolle's Theorem)[edit]

Let

Let be continuous on and differentiable on

Let

Then, there exists such that

#### Proof[edit]

If is constant, then

Hence, let be non-constant

Let such that (this is possible due to the Minimum-maximum theorem)

Without loss of generality, we can assume that

Assume if possible . Thus, and hence, there exists such that contradicting the fact that is a maximum.

Similarly, we can show that the assumption leads to a contradiction</math>. Thus,

Rolle's theorem can be generalized as follows:

## Theorem(Mean Value Theorem)[edit]

If and are differentiable on (without both having infinite derivatives at the same point) then there exists within such that

.

#### Proof[edit]

Define the function as

Obviously, this function satisfies , and by Rolle's theorem, there is a within such that .

## Theorem(Taylor's Theorem)[edit]

Let

Let be differentiable on

Then, there exists such that

#### Proof[edit]

For the proof, we use the technique known as "Telescopic Sum"

Consider the function , given by,

, where the constant is chosen so as to satisfy

By Rolle's theorem, we have that there exists such that

Expanding, we have (be careful while applying the product rule!)

Which can be rearranged to give the telescopic sum:

That is, , or

Now, we can easily see that and that

but by choice, and hence we have:

QED