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 $f(x)=x^2$
-
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.
$=\lim\limits_{h \to 0}{\frac{x^2+2xh+h^2-x^2}{h}}$ $=\lim\limits_{h \to 0}{\frac{2xh+h^2}{h}}$ -
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
$f'(x)=\lim\limits_{h \to 0}{\frac{h(2x+h)}{h}}$ $=\lim\limits_{h \to 0}(2x+h)$ -
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.
| Function | Derivative | |
|---|---|---|
| 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:
$f(x)=x^2$
$f'(x)=2x^{2-1}=2x$