Tabel turunan

Tabel turunan merupakan tabel yang menyenaraikan turunan fungsi-fungsi matematika. Operasi utama dalam kalkulus diferensial adalah mencari turunan fungsi. Dalam tabel berikut ini, f dan g adalah fungsi riil terturunkan, dan c adalah sebuah bilangan riil. Rumus-rumus berikut ini cukup untuk menurunkan fungsi elementer manapun.

Kelinearan

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

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

Kaidah darab

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

Kaidah timbalbalik

${\displaystyle \left({\frac {1}{f}}\right)'={\frac {-f'}{f^{2}}},\qquad f\neq 0}$

Kaidah hasil-bagi

${\displaystyle \left({f \over g}\right)'={f'g-fg' \over g^{2}},\qquad g\neq 0}$

Kaidah rantai

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

Turunan fungsi invers

${\displaystyle (f^{-1})'={\frac {1}{f'\circ f^{-1}}}}$

untuk setiap fungsi terdiferensialkan f dengan argumen riil dan dengan nilai riil, bila komposisi dan invers ada

Kaidah pangkat umum

${\displaystyle (f^{g})'=f^{g}\left(g'\ln f+{\frac {g}{f}}f'\right)}$

${\displaystyle c'=0\,}$

${\displaystyle x'=1\,}$

${\displaystyle (cx)'=c\,}$

${\displaystyle |x|'={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0}$

${\displaystyle (x^{c})'=cx^{c-1}\qquad {\mbox{baik }}x^{c}{\mbox{ maupun }}cx^{c-1}{\mbox{ terdefinisi}}}$

${\displaystyle \left({1 \over x}\right)'=\left(x^{-1}\right)'=-x^{-2}=-{1 \over x^{2}}}$

${\displaystyle \left({1 \over x^{c}}\right)'=\left(x^{-c}\right)'=-cx^{-(c+1)}=-{c \over x^{c+1}}}$

${\displaystyle \left({\sqrt {x}}\right)'=\left(x^{1 \over 2}\right)'={1 \over 2}x^{-{1 \over 2}}={1 \over 2{\sqrt {x}}},\qquad x>0}$

${\displaystyle \left(c^{x}\right)'=c^{x}\ln c,\qquad c>0}$

Perhatikan bahwa persamaan tersebut berlaku untuk semua c, namun turunan tersebut menghasilkan bilangan kompleks

${\displaystyle \left(e^{x}\right)'=e^{x}}$

${\displaystyle \left(^{c}\log x\right)'={\frac {1}{x\ln c}},\qquad c>0}$

${\displaystyle \left(\ln x\right)'={\frac {1}{x}}}$

${\displaystyle (\sin x)'=\cos x\,}$	${\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}$
${\displaystyle (\cos x)'=-\sin x\,}$	${\displaystyle (\arccos x)'={-1 \over {\sqrt {1-x^{2}}}}\,}$
${\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,}$	${\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}$
${\displaystyle (\sec x)'=\sec x\tan x\,}$	${\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}$
${\displaystyle (\csc x)'=-\csc x\cot x\,}$	${\displaystyle (\operatorname {arccsc} x)'={-1 \over |x|{\sqrt {x^{2}-1}}}\,}$
${\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,}$	${\displaystyle (\operatorname {arccot} x)'={-1 \over 1+x^{2}}\,}$

${\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}$	${\displaystyle (\operatorname {arcsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}$
${\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}$	${\displaystyle (\operatorname {arccosh} \,x)'={1 \over {\sqrt {x^{2}-1}}}}$
${\displaystyle (\tanh x)'=\operatorname {sech} ^{2}\,x}$	${\displaystyle (\operatorname {arctanh} \,x)'={1 \over 1-x^{2}}}$
${\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}$	${\displaystyle (\operatorname {arcsech} \,x)'={-1 \over x{\sqrt {1-x^{2}}}}}$
${\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}$	${\displaystyle (\operatorname {arccsch} \,x)'={-1 \over |x|{\sqrt {1+x^{2}}}}}$
${\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}$	${\displaystyle (\operatorname {arccoth} \,x)'={1 \over 1-x^{2}}}$