Formelsammlung Mathematik: Ableitungen

1

$(z^{\alpha }{)}^{\prime }=\alpha z^{\alpha -1}\alpha \in ℂ,z\in ℂ\setminus \{z\in ℝ:z\le 0\}$

2

$({\mathrm{e}}^z{)}^{\prime }={\mathrm{e}}^{z}z\in ℂ$

3

$(a^z{)}^{\prime }=a^z\log aa\ne 0,z\in ℂ$

4

$(\log z{)}^{\prime }=\frac{1}{z}z\in ℂ\setminus \{z\in ℝ:z\le 0\}$

5

	$\sim$	$\mathrm{\sim }\mathrm{h}$	$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{\sim }$	$\mathrm{a}\mathrm{r}\mathrm{\sim }\mathrm{h}$
$\sin$	$\cos$	$\cosh$	$\frac{1}{\sqrt{1-z^2}}$	$\frac{1}{\sqrt{1+z^2}}$
$\cos$	$-\sin$	$\sinh$	$\frac{-1}{\sqrt{1-z^2}}$	$\frac{1}{\sqrt{z-1}\sqrt{z+1}}$
$\tan$	$1+{\tan }^2$	$1-{\tanh }^2$	$\frac{1}{1+z^2}$	$\frac{1}{1-z^2}$
$\cot$	$-1-{\cot }^2$	$1-{\coth }^2$	$\frac{-1}{1+z^2}$	$\frac{1}{1-z^2}$
$\sec$	$\sec \cdot \tan$	$-\mathrm{sech}\cdot \tanh$	$\frac{1}{z^2\sqrt{1-\frac{1}{z^2}}}$	$\frac{-1}{z^2\sqrt{\frac{1}{z}-1}\sqrt{\frac{1}{z}+1}}$
$\csc$	$-\csc \cdot \cot$	$-\mathrm{csch}$	$\frac{-1}{z^2\sqrt{1-\frac{1}{z^2}}}$	$\frac{-1}{z^2\sqrt{1+\frac{1}{z^2}}}$

6

${\mathrm{\Gamma }}^{\prime }(z)=\mathrm{\Gamma }(z)\psi (z)z\notin {\mathbb{Z}}_{\le 0}$

7

$\frac{1}{2}{\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\log (x^2+y^2)=\frac{(-1{)}^{n-1}(n-1)!}{{\sqrt{x^2+y^2}}^n}\cos (n\arctan \frac{y}{x})$

8

${\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\arctan \frac{x}{y}=\frac{(-1{)}^{n-1}(n-1)!}{{\sqrt{x^2+y^2}}^n}\sin (n\arctan \frac{y}{x})$

9

${\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\frac{1}{\sqrt{1-x^2}}=\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\frac{n!(2n-2k)!}{2^{n}k!(n-k)!(n-2k)!}\frac{x^{n-2k}}{{\sqrt{1-x^2}}^{2n-2k+1}}=\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\frac{1}{2^{2k}}\frac{n{!}^2}{k{!}^2}\frac{x^{n-2k}}{(n-2k)!}\frac{1}{{\sqrt{1-x^2}}^{2n+1}}$

10

${\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\frac{1}{(1-x^2{)}^m}=\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\frac{\mathrm{\Gamma }(m+k)}{\mathrm{\Gamma }(m)}\frac{\mathrm{\Gamma }(2m+n)}{\mathrm{\Gamma }(2m+2k)}\frac{n!}{k!}\frac{x^{n-2k}}{(n-2k)!}\frac{1}{(1-x^2{)}^{n+m}}$

11

${\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n}f(x)=\sum _{k=0}^n\left\{\begin{array}{c}n\\ k\end{array}\right\}{x}^k{f}^{(k)}(x)$