# Univariate differentiation

## Partial differentiation

### The partial differential operator

#### Differential

When we change the value of an input to a function, we also change the output. We can examine these changes.

Consider the value of a function $$f(x)$$ at points $$x_1$$ and $$x_2$$.

$$y_1=f(x_1)$$

$$y_2=f(x_2)$$

$$y_2-y_1=f(x_2)-f(x_1)$$

$$\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$$

Let’s define $$x_2$$ in terms of its distance from $$x_1$$:

$$x_2=x_1+\epsilon$$

$$\dfrac{y_2-y_1}{\epsilon }=\dfrac{f(x_1+\epsilon )-f(x_1)}{\epsilon }$$

We define the differential of a function as:

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x+\epsilon )-f(x)}{\epsilon }$$

If this is defined, then we say the function is differentiable at that point.

#### Graph test

[ axis lines = left, xlabel = $$x$$, ylabel = $$f(x)$$, ] ; ;

### Differentiating constants, the identity function, and linear functions

#### Constants

$$f(x)=c$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x+\epsilon )-f(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{c-c}{\epsilon }=0$$

#### $$x$$

$$f(x)=x$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x+\epsilon )-f(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{x+\epsilon -x}{\epsilon }=1$$

$$f(x)=g(x)+h(g)$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{g(x+\epsilon )+h(x+\epsilon )-g(x)-h(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{g(x+\epsilon )-g(x)}{\epsilon }+\lim_{\epsilon \rightarrow 0^+}\dfrac{h(x+\epsilon )-h(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\dfrac{\delta g}{\delta x}+\dfrac{\delta h}{\delta x}$$

### The chain rule, the product rule and the quotient rule

#### Chain rule

$$f(x)=f(g(x))$$

$$\dfrac{\delta f}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(g(x+\epsilon) )-f(g(x))}{\epsilon }$$

$$\dfrac{\delta f}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{g(x+\epsilon )-g(x)}{g(x+\epsilon )-g(x)}\dfrac{f(g(x+\epsilon) )-f(g(x))}{\epsilon }$$

$$\dfrac{\delta f}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{g(x+\epsilon )-g(x)}{\epsilon }\dfrac{f(g(x+\epsilon) )-f(g(x))}{g(x+\epsilon )-g(x)}$$

$$\dfrac{\delta f}{\delta x}=\lim_{\epsilon \rightarrow 0^+}[\dfrac{g(x+\epsilon )-g(x)}{\epsilon }]\lim_{\epsilon \rightarrow 0^+}[\dfrac{f(g(x+\epsilon) )-f(g(x))}{g(x+\epsilon )-g(x)}]$$

$$\dfrac{\delta f}{\delta x}=\dfrac{\delta g}{\delta x}\dfrac{\delta f}{\delta g}$$

#### Product rule

$$y=f(x)g(x)$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x+\epsilon )g(x+\epsilon )-f(x)g(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x+\epsilon )g(x+\epsilon )-f(x)g(x+\epsilon )+f(x)g(x+\epsilon )-f(x)g(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x+\epsilon )g(x+\epsilon )-f(x)g(x+\epsilon )}{\epsilon }+\lim_{\epsilon \rightarrow 0^+}\dfrac{f(x)g(x+\epsilon )-f(x)g(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=\lim_{\epsilon \rightarrow 0^+}g(x+\epsilon )\dfrac{f(x+\epsilon )-f(x)}{\epsilon }+\lim_{\epsilon \rightarrow 0^+}f(x)\dfrac{g(x+\epsilon )-g(x)}{\epsilon }$$

$$\dfrac{\delta y}{\delta x}=g(x)\dfrac{\delta f}{\delta x }+f(x)\dfrac{\delta g}{\delta x}$$

#### Quotient rule

$$y=\dfrac{f(x)}{g(x)}$$

$$\dfrac{\delta }{\delta x}y=\dfrac{\delta }{\delta x}\dfrac{f(x)}{g(x)}$$

$$\dfrac{\delta }{\delta x}y=\dfrac{\delta }{\delta x}f(x)\dfrac{1}{g(x)}$$

$$\dfrac{\delta }{\delta x}y=\dfrac{\delta f}{\delta x}\dfrac{1}{g(x)}-\dfrac{\delta g}{\delta x}\dfrac{f(x)}{g(x)^2}$$

$$\dfrac{\delta }{\delta x}y=\dfrac{\dfrac{\delta f}{\delta x}g(x)-\dfrac{\delta g}{\delta x}f(x)}{g(x)^2}$$

### Differentiating natural number power functions

#### Other

$$\dfrac{\delta }{\delta x}x^n=\lim_{\delta \rightarrow 0}\dfrac{(x+\delta)^n-x^n}{\delta}$$

$$\dfrac{\delta }{\delta x}x^n=\lim_{\delta \rightarrow 0}\dfrac{(\sum_{i=0}^n x^i\delta ^{n-i}\dfrac{n!}{i!(n-i)!})-x^n}{\delta}$$

$$\dfrac{\delta }{\delta x}x^n=\lim_{\delta \rightarrow 0}\sum_{i=0}^{n-1} x^i\delta ^{n-i-1}\dfrac{n!}{i!(n-i)!}$$

$$\dfrac{\delta }{\delta x}x^n=\lim_{\delta \rightarrow 0}x^{n-1}\dfrac{n!}{(n-1)!(n-n+1)!}+\sum_{i=0}^{n-2} x^i\delta ^{n-i-1}\dfrac{n!}{i!(n-i)!}$$

$$\dfrac{\delta }{\delta x}x^n=nx^{n-1}$$

### L’Hôpital’s rule

#### L’Hôpital’s rule

If there are two functions which are both tend to $$0$$ at a limit, calculating the limit of their divisor is hard. We can use L’Hopital’s rule.

We want to calculate:

$$\lim_{x\rightarrow c}\dfrac{f(x)}{g(x)}$$

This is:

$$\lim_{x\rightarrow c}\dfrac{f(x)}{g(x)}=\lim_{x\rightarrow c}\dfrac{\dfrac{f(x)-0}{\delta}}{\dfrac{g(x)-0}{\delta}}$$

If:

$$\lim_{x\rightarrow c}f(x)=\lim_{x\rightarrow c}g(x)=0$$

Then

$$\lim_{x\rightarrow c}\dfrac{f(x)}{g(x)}=\lim_{x\rightarrow c}\dfrac{\dfrac{f(x)-f(c)}{\delta}}{\dfrac{g(x)-f(c)}{\delta}}$$

$$\lim_{x\rightarrow c}\dfrac{f(x)}{g(x)}=\dfrac{f'(x)}{g'(x)}$$

### Rolle’s theorem

#### Rolle’s theorem

Take a real function $$f(x)$$ on closed interval $$[a,b]$$, differentiable on $$(a,b,)$$, and $$f(a)=f(b)$$.

Rolle’s theorem states that:

$$\exists c\in(a,b) (f’(c)=0)$$

Generalised Rolle’s theorem states that:

Generalised Rolle’s theorem implies Rolle’s theorem, so we only need to prove the generalised theorem.

### Mean value theorem

#### Mean value theorem

Take a real function $$f(x)$$ on closed interval $$[a,b]$$, differentiable on $$(a,b,)$$.

The mean value theorem states that:

$$\exists c\in(a,b) (f’(c)=\dfrac{f(b)-f(a)}{b-a})$$

### Elasticity

#### Introduction

We have $$f(x)$$

$$Ef(x)=\dfrac{x}{f(x)}\dfrac{\delta f(x)}{\delta x}$$

This is the same as:

$$Ef(x)=\dfrac{\delta \ln f(x)}{\delta \ln x}$$

## Higher-order differentials

### Differentiable functions

#### Introduction

A differentiable function is one where the differential is defined at all points on the real line.

All differentiable functions are continuous. Not all continuous functions are differentiable.

#### Differentiability class

We can describe a function with its differentiability class. If a function can be differentiated $$n$$ times and these differentials are all continous, then the function is class $$C^n$$.

#### Smooth functions

If a function can be differentiated infinitely many times to produce continous functions, it is $$C^{\infty }$$, or smooth.

### Critial points

#### Critical points

Where partial derivative are $$0$$.