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.
Kaidah penurunan umum
sunting
c
′
=
0
{\displaystyle c'=0\,}
x
′
=
1
{\displaystyle x'=1\,}
(
c
x
)
′
=
c
{\displaystyle (cx)'=c\,}
|
x
|
′
=
x
|
x
|
=
sgn
x
,
x
≠
0
{\displaystyle |x|'={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0}
(
x
c
)
′
=
c
x
c
−
1
baik
x
c
maupun
c
x
c
−
1
terdefinisi
{\displaystyle (x^{c})'=cx^{c-1}\qquad {\mbox{baik }}x^{c}{\mbox{ maupun }}cx^{c-1}{\mbox{ terdefinisi}}}
(
1
x
)
′
=
(
x
−
1
)
′
=
−
x
−
2
=
−
1
x
2
{\displaystyle \left({1 \over x}\right)'=\left(x^{-1}\right)'=-x^{-2}=-{1 \over x^{2}}}
(
1
x
c
)
′
=
(
x
−
c
)
′
=
−
c
x
−
(
c
+
1
)
=
−
c
x
c
+
1
{\displaystyle \left({1 \over x^{c}}\right)'=\left(x^{-c}\right)'=-cx^{-(c+1)}=-{c \over x^{c+1}}}
(
x
)
′
=
(
x
1
2
)
′
=
1
2
x
−
1
2
=
1
2
x
,
x
>
0
{\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}
(
sin
x
)
′
=
cos
x
{\displaystyle (\sin x)'=\cos x\,}
(
arcsin
x
)
′
=
1
1
−
x
2
{\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}
(
cos
x
)
′
=
−
sin
x
{\displaystyle (\cos x)'=-\sin x\,}
(
arccos
x
)
′
=
−
1
1
−
x
2
{\displaystyle (\arccos x)'={-1 \over {\sqrt {1-x^{2}}}}\,}
(
tan
x
)
′
=
sec
2
x
=
1
cos
2
x
=
1
+
tan
2
x
{\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,}
(
arctan
x
)
′
=
1
1
+
x
2
{\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}
(
sec
x
)
′
=
sec
x
tan
x
{\displaystyle (\sec x)'=\sec x\tan x\,}
(
arcsec
x
)
′
=
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}
(
csc
x
)
′
=
−
csc
x
cot
x
{\displaystyle (\csc x)'=-\csc x\cot x\,}
(
arccsc
x
)
′
=
−
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arccsc} x)'={-1 \over |x|{\sqrt {x^{2}-1}}}\,}
(
cot
x
)
′
=
−
csc
2
x
=
−
1
sin
2
x
=
−
(
1
+
cot
2
x
)
{\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,}
(
arccot
x
)
′
=
−
1
1
+
x
2
{\displaystyle (\operatorname {arccot} x)'={-1 \over 1+x^{2}}\,}
(
sinh
x
)
′
=
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}
(
arcsinh
x
)
′
=
1
x
2
+
1
{\displaystyle (\operatorname {arcsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}
(
cosh
x
)
′
=
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}
(
arccosh
x
)
′
=
1
x
2
−
1
{\displaystyle (\operatorname {arccosh} \,x)'={1 \over {\sqrt {x^{2}-1}}}}
(
tanh
x
)
′
=
sech
2
x
{\displaystyle (\tanh x)'=\operatorname {sech} ^{2}\,x}
(
arctanh
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {arctanh} \,x)'={1 \over 1-x^{2}}}
(
sech
x
)
′
=
−
tanh
x
sech
x
{\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}
(
arcsech
x
)
′
=
−
1
x
1
−
x
2
{\displaystyle (\operatorname {arcsech} \,x)'={-1 \over x{\sqrt {1-x^{2}}}}}
(
csch
x
)
′
=
−
coth
x
csch
x
{\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}
(
arccsch
x
)
′
=
−
1
|
x
|
1
+
x
2
{\displaystyle (\operatorname {arccsch} \,x)'={-1 \over |x|{\sqrt {1+x^{2}}}}}
(
coth
x
)
′
=
−
csch
2
x
{\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}
(
arccoth
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {arccoth} \,x)'={1 \over 1-x^{2}}}
