# 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 ${\displaystyle a^{x}}$
${\displaystyle \mathbb {R} }$ ${\displaystyle a^{x}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle a^{x}\ln a}$ ${\displaystyle a>0}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \sin x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \cos x}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \cos x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle -\sin x}$
${\displaystyle \mathbb {R} \backslash \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)}$ ${\displaystyle \tan x}$ ${\displaystyle \mathbb {R} \backslash \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)}$ ${\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x}$
${\displaystyle \mathbb {R} \backslash \left(\pi \mathbb {Z} \right)}$ ${\displaystyle \cot x}$ ${\displaystyle \mathbb {R} \backslash \left(\pi \mathbb {Z} \right)}$ ${\displaystyle -{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x}$
${\displaystyle \left[-1,1\right]}$ ${\displaystyle \arcsin x}$ ${\displaystyle \left]-1,1\right[}$ ${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle \left[-1,1\right]}$ ${\displaystyle \arccos x}$ ${\displaystyle \left]-1,1\right[}$ ${\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \arctan x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\frac {1}{1+x^{2}}}}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {sinh} x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {cosh} x}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {cosh} x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {sinh} x}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {tanh} x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\frac {1}{\operatorname {cosh} ^{2}x}}=1-\operatorname {tanh} ^{2}x}$
${\displaystyle \mathbb {R} ^{*}}$ ${\displaystyle \operatorname {coth} x}$ ${\displaystyle \mathbb {R} ^{*}}$ ${\displaystyle {\frac {-1}{\operatorname {sinh} ^{2}x}}=1-\operatorname {coth} ^{2}x}$
${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {arsinh} \,x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\frac {1}{\sqrt {1+x^{2}}}}}$
${\displaystyle \left[1,+\infty \right[}$ ${\displaystyle \operatorname {arcosh} \,x}$ ${\displaystyle \left]1,+\infty \right[}$ ${\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}}$
${\displaystyle \left]-1,1\right[}$ ${\displaystyle \operatorname {artanh} \,x}$ ${\displaystyle \left]-1,1\right[}$ ${\displaystyle {\frac {1}{1-x^{2}}}}$

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

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