| Kalkulus | ||||||
|---|---|---|---|---|---|---|
|
||||||
|
Khusus |
||||||
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
- Kelinearan
-
(
c
f
)
′
=
c
f
′
{\displaystyle \left({cf}\right)'=cf'}
-
(
f
+
g
)
′
=
f
′
+
g
′
{\displaystyle \left({f+g}\right)'=f'+g'}
- Kaidah darab
-
(
f
g
)
′
=
f
′
g
+
f
g
′
{\displaystyle \left({fg}\right)'=f'g+fg'}
- Kaidah timbalbalik
-
(
1
f
)
′
=
−
f
′
f
2
,
f
≠
0
{\displaystyle \left({\frac {1}{f}}\right)'={\frac {-f'}{f^{2}}},\qquad f\neq 0}
- Kaidah hasil-bagi
-
(
f
g
)
′
=
f
′
g
−
f
g
′
g
2
,
g
≠
0
{\displaystyle \left({f \over g}\right)'={f'g-fg' \over g^{2}},\qquad g\neq 0}
- Kaidah rantai
-
(
f
∘
g
)
′
=
(
f
′
∘
g
)
g
′
{\displaystyle (f\circ g)'=(f'\circ g)g'}
- Turunan fungsi invers
-
(
f
−
1
)
′
=
1
f
′
∘
f
−
1
{\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
-
(
f
g
)
′
=
f
g
(
g
′
ln
f
+
g
f
f
′
)
{\displaystyle (f^{g})'=f^{g}\left(g'\ln f+{\frac {g}{f}}f'\right)}
Turunan fungsi sederhana
-
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}
Turunan fungsi eksponensial dan logaritmik
-
(
c
x
)
′
=
c
x
ln
c
,
c
>
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
-
(
e
x
)
′
=
e
x
{\displaystyle \left(e^{x}\right)'=e^{x}}
-
(
c
log
x
)
′
=
1
x
ln
c
,
c
>
0
{\displaystyle \left(^{c}\log x\right)'={\frac {1}{x\ln c}},\qquad c>0}
-
(
ln
x
)
′
=
1
x
{\displaystyle \left(\ln x\right)'={\frac {1}{x}}}
Turunan fungsi trigonometri
(
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}}\,}
 |
Turunan fungsi hiperbolik
(
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}}}
 |
Turunan fungsi khusus
|
(
Γ
(
x
)
)
′
=
∫
0
∞
t
x
−
1
e
−
t
ln
t
d
t
{\displaystyle (\Gamma (x))'=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt}
|
|
(
ζ
(
x
)
)
′
=
−
∑
n
=
1
∞
ln
n
n
x
=
−
ln
2
2
x
−
ln
3
3
x
−
ln
4
4
x
−
⋯
{\displaystyle (\zeta (x))'=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \!}
|
