Tabela de derivadas

1 Regras de derivação de funções em geral

Linearidade

$\left(cf\right)=c{f}^{\prime }$

$\left(f+g\right)=f^{\prime }+g^{\prime }$

$\left(f-g\right)=f^{\prime }-g^{\prime }$

Regra do produto

$\left(fg\right)=f^{\prime }g+f{g}^{\prime }$

Regra da divisão

$\left(\frac{f}{g}\right)=\frac{f^{\prime }g-f{g}^{\prime }}{g^2},g\ne 0$

Regra da potenciação

$(f^g{)}^{\prime }=\left(e^{g\ln f}\right)=f^g(f^{\prime }\frac{g}{f}+g^{\prime }\ln f),f>0$

Regra do encadeamento

$(f\circ g{)}^{\prime }=(f^{\prime }\circ g)g^{\prime }$

Regra do Logaritmo

$f^{\prime }=(\ln f{)}^{\prime }f,f>0$

2 Derivada de funções simples

$\frac{d}{dx}c=0$

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

$\frac{d}{dx}cx=c$

$\frac{d}{dx}|x|=\frac{|x|}{x}=\mathrm{sgn}x,x\ne 0$

$\frac{d}{dx}{x}^c=c{x}^{c-1}\text{where both }{x}^c\text{ and }c{x}^{c-1}\text{ are defined}$

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

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

$\frac{d}{dx}\sqrt{x}=\frac{d}{dx}{x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}},x>0$

3 Derivada de funções exponenciais e funções logarítmicas

$\frac{d}{dx}{c}^x=c^x\ln c,c>0$

$\frac{d}{dx}{e}^x=e^x$

$\frac{d}{dx}{\log }_{c}x=\frac{1}{x\ln c},c>0,c\ne 1$

$\frac{d}{dx}\ln x=\frac{1}{x},x>0$

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

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

4 Derivada de funções trigonométricas

$\frac{d}{dx}\sin x=\cos x$

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

$\frac{d}{dx}\tan x={\sec }^{}x=\frac{1}{{\cos }^{2}x}$

$\frac{d}{dx}\sec x=\tan x\sec x$

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

$\frac{d}{dx}\csc x=-\csc x\cot x$

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

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

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

$\frac{d}{dx}\mathrm{arcsec}x=\frac{1}{|x|\sqrt{x^2-1}}$

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

$\frac{d}{dx}\mathrm{arccsc}x=\frac{-1}{|x|\sqrt{x^2-1}}$