# Formelsammlung Mathematik: Ableitungen

##### 1
${\displaystyle (z^{\alpha })'=\alpha \,z^{\alpha -1}\qquad \alpha \in \mathbb {C} \;,\;z\in \mathbb {C} \setminus \left\{z\in \mathbb {R} \,:\,z\leq 0\right\}}$

##### 2
${\displaystyle (\mathrm {e} ^{z})'=\mathrm {e} ^{z}\qquad z\in \mathbb {C} }$

##### 3
${\displaystyle (a^{z})'=a^{z}\,\log a\qquad a\neq 0\,,\,z\in \mathbb {C} }$

##### 4
${\displaystyle (\log z)'={\frac {1}{z}}\qquad z\in \mathbb {C} \setminus \left\{z\in \mathbb {R} \,:\,z\leq 0\right\}}$

##### 5
${\displaystyle \sim }$ ${\displaystyle \operatorname {\sim \!h} }$ ${\displaystyle \operatorname {arc\!\sim } }$ ${\displaystyle \operatorname {ar\!\sim \!h} }$
${\displaystyle \sin \!}$ ${\displaystyle \cos \!}$ ${\displaystyle \cosh \!}$ ${\displaystyle {\frac {1}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {1}{\sqrt {1+z^{2}}}}}$
${\displaystyle \cos \!}$ ${\displaystyle -\sin \!}$ ${\displaystyle \sinh \!}$ ${\displaystyle {\frac {-1}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {1}{{\sqrt {z-1}}\,{\sqrt {z+1}}}}}$
${\displaystyle \tan \!}$ ${\displaystyle 1+\tan ^{2}\!}$ ${\displaystyle 1-\tanh ^{2}\!}$ ${\displaystyle {\frac {1}{1+z^{2}}}}$ ${\displaystyle {\frac {1}{1-z^{2}}}}$
${\displaystyle \cot \!}$ ${\displaystyle -1-\cot ^{2}\!}$ ${\displaystyle 1-\operatorname {coth} ^{2}\!}$ ${\displaystyle {\frac {-1}{1+z^{2}}}}$ ${\displaystyle {\frac {1}{1-z^{2}}}}$
${\displaystyle \sec \!}$ ${\displaystyle \sec \cdot \tan }$ ${\displaystyle -\operatorname {sech} \cdot \operatorname {tanh} }$ ${\displaystyle {\frac {1}{z^{2}\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}$ ${\displaystyle {\frac {-1}{z^{2}\,{\sqrt {{\frac {1}{z}}-1}}\,{\sqrt {{\frac {1}{z}}+1}}}}}$
${\displaystyle \csc \!}$ ${\displaystyle -\csc \cdot \cot }$ ${\displaystyle -\operatorname {csch} }$ ${\displaystyle {\frac {-1}{z^{2}\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}$ ${\displaystyle {\frac {-1}{z^{2}\,{\sqrt {1+{\frac {1}{z^{2}}}}}}}}$

##### 6
${\displaystyle \Gamma '(z)=\Gamma (z)\,\psi (z)\qquad z\notin \mathbb {Z} _{\leq 0}}$

##### 7
${\displaystyle {\frac {1}{2}}\left({\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}\log(x^{2}+y^{2})={\frac {(-1)^{n-1}\,(n-1)!}{{\sqrt {x^{2}+y^{2}}}^{n}}}\,\cos \left(n\,\arctan {\frac {y}{x}}\right)}$

##### 8
${\displaystyle \left({\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}\arctan {\frac {x}{y}}={\frac {(-1)^{n-1}\,(n-1)!}{{\sqrt {x^{2}+y^{2}}}^{n}}}\,\sin \left(n\,\arctan {\frac {y}{x}}\right)}$

##### 9
${\displaystyle \left({\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}{\frac {1}{\sqrt {1-x^{2}}}}=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!\,(2n-2k)!}{2^{n}\,k!\,(n-k)!\,(n-2k)!}}\,{\frac {x^{n-2k}}{{\sqrt {1-x^{2}}}^{\,2n-2k+1}}}=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {1}{2^{2k}}}\,{\frac {n!^{2}}{k!^{2}}}\,{\frac {x^{n-2k}}{(n-2k)!}}\,{\frac {1}{{\sqrt {1-x^{2}}}^{\,2n+1}}}}$

##### 10
${\displaystyle \left({\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}{\frac {1}{(1-x^{2})^{m}}}=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {\Gamma (m+k)}{\Gamma (m)}}\,{\frac {\Gamma (2m+n)}{\Gamma (2m+2k)}}\,{\frac {n!}{k!}}\,{\frac {x^{n-2k}}{(n-2k)!}}\,{\frac {1}{(1-x^{2})^{n+m}}}}$

##### 11
${\displaystyle \left(x\,{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}f(x)=\sum _{k=0}^{n}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}\,x^{k}\,f^{(k)}(x)}$