$f(x)$	$f^{\prime}(x)$	Bemerkungen
$x^n$	$n x^{n-1}$	$n=1,2,3, \ldots$
$\exp (x)$	$\exp (x)$
$a^x$	$a^x \ln (a)$	$a>0, x \in \mathbb{R}$
$\ln (\|x\|)$	$\frac{1}{x}$	$x \neq 0$
$\log (\mid x)$	$\begin{array}{l}x \\1 \\\end{array}$	$0>0 \log (x)-\ln (x)$
$\log _a(\|x\|)$	$\overline{x \ln (a)}$	$a, x>0, \log _a(x)=\frac{\ln (x)}{\ln (a)}$
$x^a$	$a x^{a-1}$	$x>0, a \in \mathbb{R}$
$\sin (x)$	$\cos (x)$
$\cos (x)$	$-\sin (x)$
$\tan (x)$	$1+\tan (x)^2=\frac{1}{\cos (x)^2}$	$x \neq \frac{\pi}{2}+k \pi, k \in \mathbf{Z}$
$\cot (x)$	$-1-\cot (x)^2=-\frac{1}{\sin (x)^2}$	$x \neq k \pi, k \in \mathbf{Z}$
$\arcsin (x)$	1	$\|x\|<1$
$\sin (x)$	$\sqrt{1-x^2}$	$\|x\|<1$
$\arccos (x)$	$\sqrt{1-x^2}$	$\|x\|<1$
$\arctan (x)$	$\frac{1}{1+x^2}$
$\text{arccot}(x)$	1
$\text{arccot}(x)$	$\overline{1+x^2}$
$\cosh (x)$	$\sinh (x)$
$\sinh (x)$	$\cosh (x)$
$\tanh (x)$	$1-\tanh (x)^2=\frac{1}{1}$
	$\overline{\cosh (x)^2}$
$\text{coth}(x)$	$1-\text{coth}(x)^2=-\frac{1}{\sinh (x)^2}$
arsinh $(x)$	1
$\text{arsinh}(x)$	$\sqrt{1+x^2}$
$\text{arcosh}(x)$	$\overline{\sqrt{x^2-1}}$	$x \in] 1, \infty[$
$\text{artgnh}(x)$	$\sqrt{x_1^2-1}$	$\|a\|<1$
$\text{artanh}(x)$	$\overline{1-x^2}$	$\|x\|<1$