Lijst van afgeleiden

Hieronder zijn f en g afleidbare functies, en c een constante.

Deze lijst bevat de afgeleiden van vele functies.

Algemene regels

\left({cf}\right)'=cf'

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

\left({f-g}\right)'=f'-g'

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

\left({f \over g}\right)'={f'g-fg' \over g^{2}}

(f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\qquad f>0

(f\circ g)'=(f'\circ g)g'

Eenvoudige functies

{d \over dx}c=0

{d \over dx}x=1

{d \over dx}|x|={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0

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

{d \over dx}{\sqrt {x}}={1 \over 2{\sqrt {x}}}

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

Exponentiële functies en logaritmische functies

{d \over dx}c^{x}={c^{x}\ln c},\qquad c>0

{d \over dx}e^{x}=e^{x}

{d \over dx}\log _{c}x={1 \over x\ln c},\qquad c>0,c\neq 1

{d \over dx}\ln x={1 \over x}

Goniometrische functies

{d \over dx}\sin x=\cos x

{d \over dx}\cos x=-\sin x

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

{d \over dx}\sec x=\sec x\tan x

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

{d \over dx}\csc x=-\cot x\csc x

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

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

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

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

{d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}}

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

Hyperbolische functies

{d \over dx}\sinh x=\cosh x

{d \over dx}\cosh x=\sinh x

{d \over dx}\tanh x={\frac {1}{{\mbox{cosh}}^{2}\,x}}

{d \over dx}{\frac {1}{\cosh x}}=-{\frac {\tanh x}{\cosh x}}

{d \over dx}\,{\mbox{coth}}\,x=-\,{\mbox{csch}}^{2}\,x

{d \over dx}\,{\mbox{csch}}\,x=-\,{\mbox{coth}}\,x\,{\mbox{csch}}\,x

{d \over dx}\sinh ^{-1}x={1 \over {\sqrt {x^{2}+1}}}

{d \over dx}\cosh ^{-1}x={1 \over {\sqrt {x^{2}-1}}}

{d \over dx}\tanh ^{-1}x={1 \over 1-x^{2}}

{d \over dx}{\mbox{sech}}^{-1}\,x={1 \over x{\sqrt {1-x^{2}}}}

{d \over dx}{\mbox{coth}}^{-1}\,x={1 \over 1-x^{2}}

{d \over dx}{\mbox{csch}}^{-1}\,x={-1 \over |x|{\sqrt {1+x^{2}}}}

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