Table de dérivées usuelles

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 ${\displaystyle x\mapsto cg'(cx)}$.

Voir aussi

Sur les autres projets Wikimedia :

• Fonction dérivée, sur Wikiversity
• Dérivation, sur Wikibooks

• Opérations sur les dérivées
• Table de primitives

• Portail de l'analyse