Cette page ou cette section ne correspond pas aux critères d'un article encyclopédique ; elle a plus sa place sur Wikiversité.

Il est recommandé de lire la page de présentation du projet avant de contacter un administrateur de Wikiversité qui procédera au transfert (procédure).

Cet article énumère les fonctions dérivées de quelques fonctions usuelles.

Domaine de définition ${\displaystyle D_{f}}$	Fonction ${\displaystyle f(x)}$	Domaine de dérivabilité ${\displaystyle D_{f'}}$	Dérivée ${\displaystyle f'(x)}$	Condition ou remarque
${\displaystyle \mathbb {R} }$	${\displaystyle k}$	${\displaystyle \mathbb {R} }$	0	${\displaystyle k}$ constante réelle
${\displaystyle \mathbb {R} }$	${\displaystyle k\,x}$	${\displaystyle \mathbb {R} }$	${\displaystyle k}$	${\displaystyle k}$ constante réelle
${\displaystyle \mathbb {R} }$	${\displaystyle x^{n}}$	${\displaystyle \mathbb {R} }$	${\displaystyle n\,x^{n-1}}$	${\displaystyle n}$ entier naturel
${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle {\frac {1}{x^{n}}}=x^{-n}}$	${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle -nx^{-n-1}=-{\frac {n}{x^{n+1}}}}$	${\displaystyle n}$ entier naturel
${\displaystyle \mathbb {R} _{+}}$	${\displaystyle {\sqrt[{n}]{x}}=x^{1/n}}$	${\displaystyle \mathbb {R} _{+}^{*}}$	${\displaystyle (1/n)x^{(1/n)-1}={\frac {1}{n{\sqrt[{n}]{x^{n-1}}}}}}$	${\displaystyle n}$ entier naturel
${\displaystyle \mathbb {R} _{+}^{*}}$	${\displaystyle x^{\alpha }}$	${\displaystyle \mathbb {R} _{+}^{*}}$	${\displaystyle \alpha x^{\alpha -1}}$	${\displaystyle \alpha }$ constante réelle. Fonction prolongeable par continuité en 0 si α ≥ 0, et de prolongée dérivable en 0 si α ≥ 1.
${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle \ln |x|}$	${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle {\frac {1}{x}}}$	Cas ${\displaystyle a=\mathrm {e} }$ de ${\displaystyle \log _{a}|x|}$
${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle \log _{a}|x|}$	${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle {\frac {1}{x\ln a}}}$	${\displaystyle a>0}$ et ${\displaystyle a\neq 1}$
${\displaystyle \mathbb {R} }$	${\displaystyle {\rm {e}}^{x}}$	${\displaystyle \mathbb {R} }$	${\displaystyle {\rm {e}}^{x}}$	Cas ${\displaystyle a=\mathrm {e} }$ de ${\displaystyle a^{x}}$
${\displaystyle \mathbb {R} }$	${\displaystyle a^{x}}$	${\displaystyle \mathbb {R} }$	${\displaystyle a^{x}\ln a}$	${\displaystyle a>0}$
${\displaystyle \mathbb {R} }$	${\displaystyle \sin x}$	${\displaystyle \mathbb {R} }$	${\displaystyle \cos x}$
${\displaystyle \mathbb {R} }$	${\displaystyle \cos x}$	${\displaystyle \mathbb {R} }$	${\displaystyle -\sin x}$
${\displaystyle \mathbb {R} \setminus \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)}$	${\displaystyle \tan x}$	${\displaystyle \mathbb {R} \setminus \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)}$	${\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x}$
${\displaystyle \mathbb {R} \setminus \left(\pi \mathbb {Z} \right)}$	${\displaystyle \cot x}$	${\displaystyle \mathbb {R} \setminus \left(\pi \mathbb {Z} \right)}$	${\displaystyle -{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x}$
${\displaystyle \left[-1,1\right]}$	${\displaystyle \arcsin x}$	${\displaystyle \left]-1,1\right[}$	${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle \left[-1,1\right]}$	${\displaystyle \arccos x}$	${\displaystyle \left]-1,1\right[}$	${\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle \mathbb {R} }$	${\displaystyle \arctan x}$	${\displaystyle \mathbb {R} }$	${\displaystyle {\frac {1}{1+x^{2}}}}$
${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {sinh} x}$	${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {cosh} x}$
${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {cosh} x}$	${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {sinh} x}$
${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {tanh} x}$	${\displaystyle \mathbb {R} }$	${\displaystyle {\frac {1}{\operatorname {cosh} ^{2}x}}=1-\operatorname {tanh} ^{2}x}$
${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle \operatorname {coth} x}$	${\displaystyle \mathbb {R} ^{*}}$	${\displaystyle {\frac {-1}{\operatorname {sinh} ^{2}x}}=1-\operatorname {coth} ^{2}x}$
${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {arsinh} \,x}$	${\displaystyle \mathbb {R} }$	${\displaystyle {\frac {1}{\sqrt {1+x^{2}}}}}$
${\displaystyle \left[1,+\infty \right[}$	${\displaystyle \operatorname {arcosh} \,x}$	${\displaystyle \left]1,+\infty \right[}$	${\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}}$
${\displaystyle \left]-1,1\right[}$	${\displaystyle \operatorname {artanh} \,x}$	${\displaystyle \left]-1,1\right[}$	${\displaystyle {\frac {1}{1-x^{2}}}}$
${\displaystyle \mathbb {R} }$	${\displaystyle {\frac {1}{1+e^{-x}}}}$	${\displaystyle \mathbb {R} }$	${\displaystyle f(x)(1-f(x))}$	Fonction sigmoïde

Si ${\displaystyle g}$ est l'une de ces fonctions, la dérivée de la fonction composée ${\displaystyle x\mapsto g(cx)}$ (où ${\displaystyle c}$ est un réel fixé) sera ${\displaystyle x\mapsto cg'(cx)}$.

• Opérations sur les dérivées
• Table de primitives

• Portail de l'analyse