Ableitungstabelle

$\frac{d}{dx}\mathrm{Const.}=0$

$\frac{d}{dx}x=1$

$\frac{d}{dx}{x}^n=n\cdot x^{n-1}$

$\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$

$\frac{d}{dx}\frac{1}{x^n}=-\frac{n}{x^{n+1}}$

$\frac{d}{dx}{a}^x=a^x\ln a$

$\frac{d}{dx}{a}^{kx}=a^{kx}k\ln a$

$\frac{d}{dx}\frac{a\cdot x+c}{b\cdot x+c}=\frac{c\cdot (a-b)}{{(b\cdot x+c)}^2}$

$\frac{d}{dx}\frac{1}{a+b\cdot x}=\frac{-b}{{(a+b\cdot x)}^2}$

${\displaystyle \frac{d}{dx}{e}^x=\left(e^x\right)\prime =e^x}$

$\frac{d}{dx}{e}^{ax}$$=\left(e^{ax}\right)\prime$$=a{e}^{ax}$

$\frac{d}{dx}{e}^{a{x}^2}$$=\left(e^{a{x}^2}\right)\prime$$=2ax{e}^{a{x}^2}$

$\frac{d}{dx}\frac{1}{e^x}$$=\left(\frac{1}{e^x}\right)\prime$$=\left(e^{-x}\right)\prime$$=-e^{-x}$$=-\frac{1}{e^x}$

$\frac{d}{dx}{e}^{\ln \left(x\right)}$$=\left(e^{\ln \left(x\right)}\right)\prime$$=\left(x\right)\prime$$=1$

$\frac{d}{dx}{e}^{x^n}$$=\left(e^{x^n}\right)\prime$$=n{x}^{n-1}{e}^{x^n}$

$\frac{d}{dx}{\left(e^x\right)}^n$$=\left({\left(e^x\right)}^n\right)\prime$$=\left(e^{nx}\right)\prime$$=n{e}^{nx}$

$\frac{d}{dx}\ln \mathrm{(}x\mathrm{)}=\frac{1}{x}$

$\frac{d}{dx}{\log }_a\mathrm{(}x\mathrm{)}=\frac{1}{x}{\log }_a\mathrm{(}e\mathrm{)}$

${\displaystyle \frac{d}{dx}\sqrt[n]{x}=\frac{d}{dx}{x}^{\frac{1}{n}}=\frac{1}{n}\cdot x^{\frac{1}{n}-1}=\frac{1}{n}\cdot x^{\frac{1-n}{n}}=\frac{1}{n}\cdot \sqrt[n]{x^{1-n}}=\frac{1}{n\cdot \sqrt[n]{x^{n-1}}}}$

${\displaystyle \frac{d}{dx}\sqrt{x}=\frac{1}{2\cdot \sqrt{x}}}$

${\displaystyle \frac{d}{dx}\sqrt[3]{x}=\frac{d}{dx}{x}^{\frac{1}{3}}=\frac{1}{3}\cdot x^{\frac{1}{3}-1}=\frac{1}{3\cdot \sqrt[3]{x^2}}}$

$\frac{d}{dx}\sin \mathrm{(}x\mathrm{)}=\cos \mathrm{(}x\mathrm{)}$

$\frac{d}{dx}\cos \mathrm{(}x\mathrm{)}=-\sin \mathrm{(}x\mathrm{)}$

$\frac{d}{dx}\sin \mathrm{(}kx\mathrm{)}$$=k\cos \mathrm{(}kx\mathrm{)}$

$\frac{d}{dx}\cos \mathrm{(}kx\mathrm{)}$$=-k\sin \mathrm{(}kx\mathrm{)}$

$\frac{d}{dx}\tan \mathrm{(}x\mathrm{)}$$=\frac{d}{dx}\frac{\sin \mathrm{(}x\mathrm{)}}{\cos \mathrm{(}x\mathrm{)}}$$=\frac{1}{{\cos }^2\mathrm{(}x\mathrm{)}}$

$\frac{d}{dx}\cot \mathrm{(}x\mathrm{)}=-\frac{1}{{\sin }^2\mathrm{(}x\mathrm{)}}$

$\frac{d}{dx}\arcsin \mathrm{(}x\mathrm{)}=\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\arccos \mathrm{(}x\mathrm{)}=-\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\arctan \mathrm{(}x\mathrm{)}=\frac{1}{1+x^2}$

$\frac{d}{dx}\mathrm{arccot}\mathrm{(}x\mathrm{)}=-\frac{1}{1+x^2}$

$\frac{d}{dx}\sinh \mathrm{(}x\mathrm{)}=\cosh \mathrm{(}x\mathrm{)}$

$\frac{d}{dx}\cosh \mathrm{(}x\mathrm{)}=\sinh \mathrm{(}x\mathrm{)}$

$\frac{d}{dx}\tanh \mathrm{(}x\mathrm{)}=\frac{1}{{\cosh }^2\mathrm{(}x\mathrm{)}}$

$\frac{d}{dx}\coth \mathrm{(}x\mathrm{)}=-\frac{1}{{\sinh }^2\mathrm{(}x\mathrm{)}}$

$\frac{d}{dx}\mathrm{arsinh}\mathrm{(}x\mathrm{)}=\frac{1}{\sqrt{1+x^2}}$

$\frac{d}{dx}\mathrm{arcosh}\mathrm{(}x\mathrm{)}=\frac{1}{\sqrt{x^2-1}}$

$\frac{d}{dx}\mathrm{artanh}\mathrm{(}x\mathrm{)}=\frac{1}{1-x^2}$

$\frac{d}{dx}\mathrm{arcoth}\mathrm{(}x\mathrm{)}=-\frac{1}{1-x^2}$

$\frac{d^n}{d{x}^n}{a}^x=a^x{\left(\ln a\right)}^n$

$\frac{d^n}{d{x}^n}{a}^{kx}=a^{kx}{\left(k\ln a\right)}^n$

$\frac{d^n}{d{x}^n}\sin \mathrm{(}x\mathrm{)}=\sin \mathrm{(}x+\frac{n\pi }{2}\mathrm{)}$

$\frac{d^n}{d{x}^n}\cos \mathrm{(}x\mathrm{)}=\cos \mathrm{(}x+\frac{n\pi }{2}\mathrm{)}$

$\frac{d^n}{d{x}^n}\sin \mathrm{(}kx\mathrm{)}=k^n\sin \mathrm{(}kx+\frac{n\pi }{2}\mathrm{)}$

$\frac{d^n}{d{x}^n}\cos \mathrm{(}kx\mathrm{)}=k^n\cos \mathrm{(}kx+\frac{n\pi }{2}\mathrm{)}$