# Table of derivatives

Basic derivatives:

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

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

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

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

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

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

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

Derivatives of fractions:

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

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

Derivatives of the e-functions:

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

$\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}^{\mathrm{ln}\left(x\right)}$$=\left({e}^{\mathrm{ln}\left(x\right)}\right)\prime$$=\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}$

Derivatives of logarithm functions:

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

$\frac{d}{dx}{\mathrm{log}}_{a}\left(x\right)=\frac{1}{x}{\mathrm{log}}_{a}\left(e\right)$

Derivatives of root functions:

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

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

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

Derivatives of trigonometric functions:

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

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

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

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

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

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

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

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

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

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

Derivatives of hyperbolic functions and area functions:

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

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

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

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

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

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

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

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

n-th derivative:

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

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

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

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

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

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

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