Tabela de derivadas

Regra da soma

• ${\displaystyle \left({f+g}\right)'=f'+g'}$

Regra da subtração

• ${\displaystyle (f-g)'=f'-g'}$

Regra da multiplicação

• ${\displaystyle (cf)'=cf'}$

Regra do produto

• ${\displaystyle \left({fg}\right)'=f'g+fg'}$

Regra do quociente

• ${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}$ sendo esta válida para todo ${\displaystyle x}$ no domínio das funções com ${\displaystyle g(x)\neq 0}$.

Regra da Cadeia

• ${\displaystyle (f\circ g)'(x)=f'{\big (}g(x){\big )}g'(x)}$

onde ${\displaystyle (f\circ g)(x):=f{\big (}g(x){\big )}}$ é a composição de ${\displaystyle f}$ com ${\displaystyle g}$ (usualmente, lê-se "${\displaystyle f}$ após ${\displaystyle g}$"). Esta é válida para ${\displaystyle x}$ no domínio ${\displaystyle D_{g}}$ da função ${\displaystyle g}$ e tal que ${\displaystyle g(x)}$ esteja no domínio ${\displaystyle D_{f}}$ da função ${\displaystyle f}$, ou seja, é válida em ${\displaystyle D_{f\circ g}=\{x\in D_{g}:g(x)\in D_{f}\}}$.

• ${\displaystyle {d \over dx}c=0}$

• ${\displaystyle {d \over dx}x=1}$

• ${\displaystyle {d \over dx}c.x=c}$

• ${\displaystyle {d \over dx}x^{c}=cx^{c-1}}$

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

• ${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}$
• ${\displaystyle {\frac {d}{dx}}\log _{b}|x|={\frac {1}{x\ln b}}}$
• ${\displaystyle {\frac {d}{dx}}\ln |x|={\frac {1}{x}}}$

Se ${\displaystyle u}$ é uma função derivável, então:

• ${\displaystyle {\frac {d}{dx}}e^{u}=u'e^{u}}$

• ${\displaystyle {\frac {d}{dx}}a^{u}=u'a^{u}\ln a}$

• ${\displaystyle {\frac {d}{dx}}\log _{a}u={\frac {u'}{u}}\log _{a}e}$

• ${\displaystyle {\frac {d}{dx}}\ln |u|={\frac {u'}{u}}}$

Função	Abreviatura	Identidade trigonométrica
Seno	sen (ou sin)	${\displaystyle \operatorname {sen} \theta \equiv \cos \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\csc \theta }}}$
Cosseno	cos	${\displaystyle \cos \theta \equiv \operatorname {sen} \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\sec \theta }}}$
Tangente	tan (ou tg)	${\displaystyle \tan \theta \equiv {\frac {\operatorname {sen} \theta }{\cos \theta }}\equiv \cot \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\cot \theta }}}$
Cossecante	csc (ou cosec)	${\displaystyle \csc \theta \equiv \sec \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\operatorname {sen} \theta }}}$
Secante	sec	${\displaystyle \sec \theta \equiv \csc \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\cos \theta }}}$
Cotangente	cot (ou cotg ou cotan)	${\displaystyle \cot \theta \equiv {\frac {\cos \theta }{\operatorname {sen} \theta }}\equiv \tan \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\tan \theta }}}$

• ${\displaystyle {\frac {d}{dx}}\operatorname {sen} x=\cos x}$

• ${\displaystyle {\frac {d}{dx}}\cos x=-\operatorname {sen} x}$

• ${\displaystyle {\frac {d}{dx}}\tan x=\sec ^{2}x}$

• ${\displaystyle {d \over dx}\csc x=-\csc x\cot x}$

• ${\displaystyle {d \over dx}\sec x=\sec x\tan x}$

• ${\displaystyle {d \over dx}\cot x=-\csc ^{2}x}$

• ${\displaystyle {d \over dx}\operatorname {arcsen} x={\frac {1}{\sqrt {1-x^{2}}}}}$

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

• ${\displaystyle {d \over dx}\arctan \,x={\frac {1}{1+x^{2}}}}$

• ${\displaystyle {d \over dx}\operatorname {arcsec} x={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$

• ${\displaystyle {d \over dx}\operatorname {arccot} x=-{\frac {1}{1+x^{2}}}}$

• ${\displaystyle {d \over dx}\operatorname {arccsc} x=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$

• ${\displaystyle {d \over dx}\operatorname {senh} x=\cosh x}$

• ${\displaystyle {d \over dx}\cosh x=\operatorname {senh} x}$

• ${\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}x}$

• ${\displaystyle {d \over dx}\operatorname {sech} x=-\operatorname {sech} x\tanh x}$

• ${\displaystyle {d \over dx}\operatorname {coth} x=-\operatorname {csch} ^{2}x}$

• ${\displaystyle {d \over dx}\,\operatorname {csch} x=-\operatorname {csch} x\operatorname {coth} x}$

• ${\displaystyle {d \over dx}\operatorname {argsenh} x={\frac {1}{\sqrt {x^{2}+1}}}}$

• ${\displaystyle {d \over dx}\operatorname {argcosh} x={\frac {1}{\sqrt {x^{2}-1}}}}$

• ${\displaystyle {d \over dx}\operatorname {argtanh} x={\frac {1}{1-x^{2}}}}$

• ${\displaystyle {d \over dx}\operatorname {argsech} x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}}$

• ${\displaystyle {d \over dx}\operatorname {argcoth} x={\frac {1}{1-x^{2}}}}$

• ${\displaystyle {d \over dx}\operatorname {argcsch} x=-{1 \over |x|{\sqrt {x^{2}+1}}}}$

• Tabela de integrais