Ableitungen

Common derivatives - Ableitungen spezieller Funktionen

$${f(x)}$$	$${f'(x)}$$	$${f^{\prime\prime}(x)}$$
a = const.	0	0
$${x^n}$$	$${nx^{n-1}}$$	$${n(n-1)x^{n-2}}$$
$${\sqrt{x}}$$	$${\frac{1}{2 \sqrt{x}}}$$	$${-\frac{1}{4 \sqrt{x}}}$$
$${a^x}$$	$${a^{x} \ln{a}}$$	$${a^{x} (\ln{a})^2}$$
$${e^x}$$	$${e^{x}}$$	$${e^{x}}$$
$${\sin{x}}$$	$${\cos{x}}$$	$${-\sin{x}}$$
$${\cos{x}}$$	$${-\sin{x}}$$	$${-\cos{x}}$$
$${\tan{x}}$$	$${\sec^2{x} = \frac{1}{\cos^2{x}}= 1+\tan^2{x}}$$	$${2\tan{x}(1+\tan^2{x})}$$
$${\log_{a}{x}}$$	$${\frac{1}{x \cdot \log_{a}{x}}}$$	$${\frac{-1}{x^2 \cdot \log_{a}{x}}}$$
$${\ln{x}}$$	$${\frac{1}{x}}$$	$${-\frac{1}{x^2}}$$

chain rule - Kettenregel

$${[f(g(x))]' = f'(g(x)) \cdot g'(x)}$$

product rule - Produktregel

$${[f(x) \cdot g(x)]' = f'(x) \cdot g(x) + f(x) \cdot g'(x)}$$

quotient rule - Quotientenregel

$${\left[\frac{f(x)}{g(x)}\right]' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}}$$