Math Differentiability and derivative Differentiate/Derive

# Deriving/Differentiating

Calculating the derivative function is called differentiating or deriving.
There are two important methods for differentiating.

## h-Method

The h-method uses the differential quotient. You do not insert a specific point, instead the distance of the points ($h=x-x_0$) goes towards 0:

$f'(x)=\lim\limits_{h \to 0}{\frac{f(x+h)-f(x)}{h}}$

### Example

Differentiating $f(x)=x^2$

1. #### Inserting

$f'(x)=\lim\limits_{h \to 0}{\frac{f(x+h)-f(x)}{h}}$
$f'(x)=\lim\limits_{h \to 0}{\frac{(x+h)^2-x^2}{h}}$

Applying binomial formulas and simplifying.
$=\lim\limits_{h \to 0}{\frac{x^2+2xh+h^2-x^2}{h}}$ $=\lim\limits_{h \to 0}{\frac{2xh+h^2}{h}}$
2. #### Reducing

Now the fraction has to be simplified as, inserting 0 for h, the denominator would be 0 (division through 0 not allowed!).
$f'(x)=\lim\limits_{h \to 0}{\frac{2xh+h^2}{h}}$
Reducing
$f'(x)=\lim\limits_{h \to 0}{\frac{h(2x+h)}{h}}$ $=\lim\limits_{h \to 0}(2x+h)$
3. #### inserting $h=0$

Now $h$ runs towards 0and gives the derivative.
$f'(x)=\lim\limits_{h \to 0}(2x+\overbrace{h}^{\to 0})=2x$

## Differentiating using derivative rules

A much easier method to differentiate a function is to use the derivative rules.

FunctionDerivative
Constant, power and factor rule
$f(x)=c$ $f'(x)=0$
$f(x)=x^n$ $f'(x)=n\cdot x^{n-1}$
$f(x)=k\cdot g(x)$ $f'(x)=k\cdot g'(x)$
Sum rule
$f(x)=g(x)+h(x)$ $f'(x)=g'(x)+h'(x)$
Product rule
$f(x)=g(x)\cdot h(x)$ $f'(x)=g'(x)\cdot h(x)$ $+h'(x)\cdot g(x)$
Quotient rule
$f(x)=\frac{g(x)}{h(x)}$ $f'(x)=\frac{g'(x)\cdot h(x) -h'(x)\cdot g(x)}{(h(x))^2}$
Chain rule
$f(x)=g(h(x))$ $f'(x)=g'(h(x))\cdot h'(x)$

### Example

Power rule:
$f(x)=x^2$
$f'(x)=2x^{2-1}=2x$