Derivatives
Definition
$$f'(x)= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Formula
- $c$ is a constant
- $f$, $g$ are functions of a variable $x$
Basic properties
- Linearity
$$(cf+g)'=cf'+g'$$
- Product
$$(f.g)'=f'.g + g'.f$$
- Inverse (Quotient with $f=1$)
$$(\frac{1}{g})'=\frac{-g'}{g^2}$$
- Quotient
$$(\frac{f}{g})'=\frac{f'.g-g'.f}{g^2}$$
- Chain
$$(g \circ f)'=(g' \circ f) . f'$$
Usual functions
| $f(x)$ | Domain $I$ | $f'(x)$ |
|---|---|---|
| $c$ | $\Bbb{R}$ | $0$ |
| $x$ | $\Bbb{R}$ | $1$ |
| $x^n$ ($n\in\Bbb{N^*}$) | $\Bbb{R}$ | $nx^{n-1}$ |
| $\ln x$ | $]0,+\infty[$ | $1/x$ |
| $e^x$ | $\Bbb{R}$ | $e^x$ |
| $\cos x$ | $\Bbb{R}$ | $-\sin x$ |
| $\sin x$ | $\Bbb{R}$ | $\cos x$ |
| $\tan x$ | $]-\pi/2,+\pi/2[$ | $1+\tan^2 x = 1/\cos^2 x$ |
| $\cosh x$ | $\Bbb{R}$ | $\sinh x$ |
| $\sinh x$ | $\Bbb{R}$ | $\cosh x$ |
| $\tanh x$ | $\Bbb{R}$ | $1/\cosh^2 x = 1 - \tanh^2 x$ |