以下的列表列出了许多函数的导数。f 和g是可微函数，而别的皆为常数。用这些公式，可以求出任何初等函数的导数。
目录
1 一般求导法则
2 代数函数的导数
3 指数和对数函数的导数
4 三角函数的导数
5 反三角函數的導數
6 双曲函数的导数
7 特殊函数的导数
8 註釋
一般求导法则
编辑
線性法则
d
(
M
f
)
d
x
=
M
d
f
d
x
;
[
M
f
(
x
)
]
′
=
M
f
′
(
x
)
{\displaystyle {{\mbox{d}}(Mf) \over {\mbox{d}}x}=M{{\mbox{d}}f \over {\mbox{d}}x};\qquad [Mf(x)]'=Mf'(x)}
d
(
f
±
g
)
d
x
=
d
f
d
x
±
d
g
d
x
{\displaystyle {{{\mbox{d}}(f\pm g)} \over {{\mbox{d}}x}}={{\mbox{d}}f \over {\mbox{d}}x}\pm {{\mbox{d}}g \over {\mbox{d}}x}\ }
乘法定则
d
f
g
d
x
=
d
f
d
x
g
+
f
d
g
d
x
{\displaystyle {{\mbox{d}}fg \over {\mbox{d}}x}={{\mbox{d}}f \over {\mbox{d}}x}g+f{\frac {{\mbox{d}}g}{{\mbox{d}}x}}}
除法定则
d
f
g
d
x
=
d
f
d
x
g
−
f
d
g
d
x
g
2
(
g
≠
0
)
{\displaystyle {\frac {{\mbox{d}}{\dfrac {f}{g}}}{{\mbox{d}}x}}={\frac {{\dfrac {{\mbox{d}}f}{{\mbox{d}}x}}g-f{\dfrac {{\mbox{d}}g}{{\mbox{d}}x}}}{g^{2}}}\qquad (g\neq 0)}
倒数定则
d
1
g
d
x
=
−
d
g
d
x
g
2
(
g
≠
0
)
{\displaystyle {\frac {{\mbox{d}}{\dfrac {1}{g}}}{{\mbox{d}}x}}={\frac {-{\dfrac {{\mbox{d}}g}{{\mbox{d}}x}}}{g^{2}}}\qquad (g\neq 0)}
复合函数求导法则（連鎖定則）
(
f
∘
g
)
′
(
x
)
=
f
′
(
g
(
x
)
)
g
′
(
x
)
.
{\displaystyle (f\circ g)'(x)=f'(g(x))g'(x).}
d
f
[
g
(
x
)
]
d
x
=
d
f
(
g
)
d
g
d
g
d
x
=
f
′
[
g
(
x
)
]
g
′
(
x
)
{\displaystyle {\frac {{\mbox{d}}f[g(x)]}{{\mbox{d}}x}}={\frac {{\mbox{d}}f(g)}{{\mbox{d}}g}}{\frac {{\mbox{d}}g}{{\mbox{d}}x}}=f'[g(x)]g'(x)}
反函数的导数
由于
g
(
f
(
x
)
)
=
x
{\displaystyle g(f(x))=x}
，故
g
(
f
(
x
)
)
′
=
1
{\displaystyle g(f(x))'=1}
，根據复合函数求导法则，則
g
(
f
(
x
)
)
′
=
d
g
[
f
(
x
)
]
d
x
=
d
g
(
f
)
d
f
d
f
d
x
=
1
{\displaystyle g(f(x))'={\frac {{\mbox{d}}g[f(x)]}{{\mbox{d}}x}}={\frac {{\mbox{d}}g(f)}{{\mbox{d}}f}}{\frac {{\mbox{d}}f}{{\mbox{d}}x}}=1}
所以
d
f
d
x
=
1
d
g
(
f
)
d
f
=
[
d
g
(
f
)
d
f
]
−
1
=
[
g
′
(
f
)
]
−
1
{\displaystyle {\frac {{\mbox{d}}f}{{\mbox{d}}x}}={\frac {1}{\dfrac {{\mbox{d}}g(f)}{{\mbox{d}}f}}}=[{\frac {{\mbox{d}}g(f)}{{\mbox{d}}f}}]^{-1}=[g'(f)]^{-1}}
同理
d
g
d
x
=
1
d
f
(
g
)
d
g
=
[
d
f
(
g
)
d
g
]
−
1
=
[
f
′
(
g
)
]
−
1
{\displaystyle {\frac {{\mbox{d}}g}{{\mbox{d}}x}}={\frac {1}{\dfrac {{\mbox{d}}f(g)}{{\mbox{d}}g}}}=[{\frac {{\mbox{d}}f(g)}{{\mbox{d}}g}}]^{-1}=[f'(g)]^{-1}}
广义幂法则
(
f
g
)
′
=
(
e
g
ln
f
)
′
=
f
g
(
g
′
ln
f
+
g
f
f
′
)
{\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(g'\ln f+{\frac {g}{f}}f'\right)}
代数函数的导数
编辑
(n为任意实常数)
d
n
d
x
=
0
{\displaystyle {{\mbox{d}}n \over {\mbox{d}}x}=0}
d
x
d
x
=
1
{\displaystyle {{\mbox{d}}x \over {\mbox{d}}x}=1}
d
x
n
d
x
=
n
x
n
−
1
{\displaystyle {{\mbox{d}}x^{n} \over {\mbox{d}}x}=nx^{n-1}\qquad }
當
n
≤
1
{\displaystyle n\leq 1}
，則
x
≠
0
{\displaystyle x\neq 0}
d
|
x
|
d
x
=
x
|
x
|
=
|
x
|
x
=
sgn
x
x
≠
0
{\displaystyle {{\mbox{d}}|x| \over {\mbox{d}}x}={x \over |x|}={|x| \over x}=\operatorname {sgn} x\qquad x\neq 0}
指数和对数函数的导数
编辑
d
e
f
(
x
)
d
x
=
f
′
(
x
)
e
f
(
x
)
{\displaystyle {\frac {{\mbox{d}}\ e^{f(x)}}{{\mbox{d}}x}}=f'(x)e^{f(x)}}
d
e
x
d
x
=
lim
Δ
x
→
0
e
x
−
e
x
−
Δ
x
Δ
x
=
e
x
lim
Δ
x
→
0
1
−
e
−
Δ
x
Δ
x
=
e
x
{\displaystyle {\begin{aligned}{\frac {{\mbox{d}}\ e^{x}}{{\mbox{d}}x}}&=\lim _{\Delta x\to 0}{\frac {e^{x}-e^{x-\Delta x}}{\Delta x}}\\&=e^{x}\lim _{\Delta x\to 0}{\frac {1-e^{-\Delta x}}{\Delta x}}\\&=e^{x}\end{aligned}}}
d
α
x
d
x
=
d
e
x
ln
α
d
x
=
d
e
x
ln
α
d
x
ln
α
⋅
d
x
ln
α
d
x
=
e
x
ln
α
ln
α
=
α
x
ln
α
{\displaystyle {\begin{aligned}{\frac {{\mbox{d}}\ \alpha ^{x}}{{\mbox{d}}x}}&={\frac {{\mbox{d}}\ e^{x\!\ln \!\alpha }}{{\mbox{d}}x}}\\&={\frac {{\mbox{d}}e^{x\!\ln \!\alpha }}{{\mbox{d}}\ x\!\ln \!\alpha }}\cdot {\frac {{\mbox{d}}\ x\!\ln \!\alpha }{{\mbox{d}}x}}\\&=e^{x\!\ln \!\alpha }\!\ln \!\alpha \\&=\alpha ^{x}\!\ln \!\alpha \end{aligned}}}
[註 1]
d
ln
x
d
x
=
lim
h
→
0
ln
(
x
+
h
)
−
ln
x
h
=
lim
h
→
0
(
1
h
ln
(
x
+
h
x
)
)
=
lim
h
→
0
(
x
x
h
ln
(
1
+
h
x
)
)
=
1
x
ln
(
lim
h
→
0
(
1
+
h
x
)
x
h
)
=
1
x
ln
e
=
1
x
{\displaystyle {\begin{aligned}{\frac {{\mbox{d}}\ln x}{{\mbox{d}}x}}&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}({\frac {1}{h}}\ln({\frac {x+h}{x}}))\\&=\lim _{h\to 0}({\frac {x}{xh}}\ln(1+{\frac {h}{x}}))\\&={\frac {1}{x}}\ln(\lim _{h\to 0}(1+{\frac {h}{x}})^{\frac {x}{h}})\\&={\frac {1}{x}}\ln e\\&={\frac {1}{x}}\end{aligned}}}
d
log
α
|
x
|
d
x
=
1
ln
α
d
ln
|
x
|
d
x
=
1
x
ln
α
{\displaystyle {\frac {{\mbox{d}}\log _{\alpha }|x|}{{\mbox{d}}x}}={1 \over \ln \alpha }{\frac {{\mbox{d}}\ln |x|}{{\mbox{d}}x}}={1 \over x\ln \alpha }}
d
x
x
d
x
=
x
x
(
1
+
ln
x
)
{\displaystyle {\frac {{\mbox{d}}\ x^{x}}{{\mbox{d}}x}}=x^{x}(1+\ln x)}
[註 2]
三角函数的导数
编辑
(
sin
x
)
′
=
lim
h
→
0
sin
(
x
+
h
)
−
sin
x
h
=
lim
h
→
0
sin
x
cos
h
+
cos
x
sin
h
−
sin
x
h
=
lim
h
→
0
(
sin
x
cos
h
−
1
h
+
cos
x
sin
h
h
)
=
cos
x
{\displaystyle {\begin{aligned}(\sin x)'&=\lim _{h\to 0}{\frac {\sin(x+h)-\sin x}{h}}\\&=\lim _{h\to 0}{\frac {\sin x\cos h+\cos x\sin h-\sin x}{h}}\\&=\lim _{h\to 0}(\sin x{\frac {\cos h-1}{h}}+\cos x{\frac {\sin h}{h}})\\&=\cos x\end{aligned}}}
(
cos
x
)
′
=
lim
h
→
0
cos
(
x
+
h
)
−
cos
x
h
=
lim
h
→
0
cos
x
cos
h
−
sin
x
sin
h
−
cos
x
h
=
lim
h
→
0
(
cos
x
cos
h
−
1
h
−
sin
x
sin
h
h
)
=
−
sin
x
{\displaystyle {\begin{aligned}(\cos x)'&=\lim _{h\to 0}{\frac {\cos(x+h)-\cos x}{h}}\\&=\lim _{h\to 0}{\frac {\cos x\cos h-\sin x\sin h-\cos x}{h}}\\&=\lim _{h\to 0}(\cos x{\frac {\cos h-1}{h}}-\sin x{\frac {\sin h}{h}})\\&=-\sin x\end{aligned}}}
(
tan
x
)
′
=
(
sin
x
cos
x
)
′
=
(
sin
x
)
′
cos
x
−
sin
x
(
cos
x
)
′
cos
2
x
=
cos
2
x
+
sin
2
x
cos
2
x
=
1
cos
2
x
=
sec
2
x
{\displaystyle {\begin{aligned}(\tan x)'&=({\frac {\sin x}{\cos x}})'\\&={\frac {(\sin x)'\cos x-\sin x(\cos x)'}{\cos ^{2}x}}\\&={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}\\&={\frac {1}{\cos ^{2}x}}=\sec ^{2}x\end{aligned}}}
(
cot
x
)
′
=
(
cos
x
sin
x
)
′
=
(
cos
x
)
′
sin
x
−
cos
x
(
sin
x
)
′
sin
2
x
=
−
sin
2
x
−
cos
2
x
sin
2
x
=
−
1
sin
2
x
=
−
csc
2
x
{\displaystyle {\begin{aligned}(\cot x)'&=({\frac {\cos x}{\sin x}})'\\&={\frac {(\cos x)'\sin x-\cos x(\sin x)'}{\sin ^{2}x}}\\&={\frac {-\sin ^{2}x-\cos ^{2}x}{\sin ^{2}x}}\\&=-{\frac {1}{\sin ^{2}x}}=-\csc ^{2}x\end{aligned}}}
(
sec
x
)
′
=
(
1
cos
x
)
′
=
sin
x
cos
2
x
=
sec
x
tan
x
{\displaystyle {\begin{aligned}(\sec x)'&=({\frac {1}{\cos x}})'\\&={\frac {\sin x}{\cos ^{2}x}}\\&=\sec x\tan x\end{aligned}}}
(
csc
x
)
′
=
(
1
sin
x
)
′
=
−
cos
x
sin
2
x
=
−
csc
x
cot
x
{\displaystyle {\begin{aligned}(\csc x)'&=({\frac {1}{\sin x}})'\\&={\frac {-\cos x}{\sin ^{2}x}}\\&=-\csc x\cot x\end{aligned}}}
反三角函數的導數
编辑
(
arcsin
x
)
′
=
1
cos
(
arcsin
x
)
⇔
sin
(
arcsin
x
)
=
x
⇔
cos
(
arcsin
x
)
(
arcsin
x
)
′
=
1
=
1
1
−
sin
2
(
arcsin
x
)
=
1
1
−
x
2
(
|
x
|
<
1
)
{\displaystyle {\begin{aligned}(\arcsin x)'&={\frac {1}{\cos(\arcsin x)}}\Leftrightarrow \sin(\arcsin x)=x\Leftrightarrow \cos(\arcsin x)(\arcsin x)'=1\\&={\frac {1}{\sqrt {1-\sin ^{2}(\arcsin x)}}}\\&={\frac {1}{\sqrt {1-x^{2}}}}\ \ (\left|x\right|<1)\end{aligned}}}
(
arccos
x
)
′
=
1
−
sin
(
arccos
x
)
⇔
cos
(
arccos
x
)
=
x
⇔
−
sin
(
arccos
x
)
(
arccos
x
)
′
=
1
=
−
1
1
−
cos
2
(
arccos
x
)
=
−
1
1
−
x
2
(
|
x
|
<
1
)
{\displaystyle {\begin{aligned}(\arccos x)'&={\frac {1}{-\sin(\arccos x)}}\Leftrightarrow \cos(\arccos x)=x\Leftrightarrow -\sin(\arccos x)(\arccos x)'=1\\&=-{\frac {1}{\sqrt {1-\cos ^{2}(\arccos x)}}}\\&=-{\frac {1}{\sqrt {1-x^{2}}}}\ \ (\left|x\right|<1)\end{aligned}}}
(
arctan
x
)
′
=
1
sec
2
(
arctan
x
)
⇔
tan
(
arctan
x
)
=
x
⇔
sec
2
(
arctan
x
)
(
arctan
x
)
′
=
1
=
1
1
+
tan
2
(
arctan
x
)
=
1
1
+
x
2
{\displaystyle {\begin{aligned}(\arctan x)'&={\frac {1}{\sec ^{2}(\arctan x)}}\Leftrightarrow \tan(\arctan x)=x\Leftrightarrow \sec ^{2}(\arctan x)(\arctan x)'=1\\&={\frac {1}{1+\tan ^{2}(\arctan x)}}\\&={\frac {1}{1+x^{2}}}\end{aligned}}}
(
arccot
x
)
′
=
1
−
csc
2
(
arccot
x
)
⇔
cot
(
arccot
x
)
=
x
⇔
−
csc
2
(
arccot
x
)
(
arccot
x
)
′
=
1
=
−
1
1
+
cot
2
(
arccot
x
)
=
−
1
1
+
x
2
{\displaystyle {\begin{aligned}(\operatorname {arccot} x)'&={\frac {1}{-\csc ^{2}(\operatorname {arccot} x)}}\Leftrightarrow \cot(\operatorname {arccot} x)=x\Leftrightarrow -\csc ^{2}(\operatorname {arccot} x)(\operatorname {arccot} x)'=1\\&=-{\frac {1}{1+\cot ^{2}(\operatorname {arccot} x)}}\\&=-{\frac {1}{1+x^{2}}}\end{aligned}}}
(
arcsec
x
)
′
=
1
sec
(
arcsec
x
)
tan
(
arcsec
x
)
⇔
sec
(
arcsec
x
)
=
x
⇔
sec
(
arcsec
x
)
tan
(
arcsec
x
)
(
arcsec
x
)
′
=
1
=
1
|
x
|
sec
2
(
arcsec
x
)
−
1
=
1
|
x
|
x
2
−
1
(
|
x
|
>
1
)
{\displaystyle {\begin{aligned}(\operatorname {arcsec} x)'&={\frac {1}{\sec(\operatorname {arcsec} x)\tan(\operatorname {arcsec} x)}}\Leftrightarrow \sec(\operatorname {arcsec} x)=x\Leftrightarrow \sec(\operatorname {arcsec} x)\tan(\operatorname {arcsec} x)(\operatorname {arcsec} x)'=1\\&={\frac {1}{|x|{\sqrt {\sec ^{2}(\operatorname {arcsec} x)-1}}}}\\&={\frac {1}{|x|{\sqrt {x^{2}-1}}}}\ \ (\left|x\right|>1)\end{aligned}}}
(
arccsc
x
)
′
=
1
−
csc
(
arccsc
x
)
cot
(
arccsc
x
)
⇔
csc
(
arccsc
x
)
=
x
⇔
−
csc
(
arccsc
x
)
cot
(
arccsc
x
)
(
arccsc
x
)
′
=
1
=
−
1
|
x
|
csc
2
(
arcsec
x
)
−
1
=
−
1
|
x
|
x
2
−
1
(
|
x
|
>
1
)
{\displaystyle {\begin{aligned}(\operatorname {arccsc} x)'&={\frac {1}{-\csc(\operatorname {arccsc} x)\cot(\operatorname {arccsc} x)}}\Leftrightarrow \csc(\operatorname {arccsc} x)=x\Leftrightarrow -\csc(\operatorname {arccsc} x)\cot(\operatorname {arccsc} x)(\operatorname {arccsc} x)'=1\\&=-{\frac {1}{|x|{\sqrt {\csc ^{2}(\operatorname {arcsec} x)-1}}}}\\&=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}\ \ (\left|x\right|>1)\end{aligned}}}
双曲函数的导数
编辑
(
sinh
x
)
′
=
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}
(
arsinh
x
)
′
=
1
x
2
+
1
{\displaystyle (\operatorname {arsinh} \,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}}}
(
arcosh
x
)
′
=
1
x
2
−
1
(
x
>
1
)
{\displaystyle (\operatorname {arcosh} \,x)'={1 \over {\sqrt {x^{2}-1}}}(x>1)}
(
tanh
x
)
′
=
sech
2
x
{\displaystyle (\tanh x)'=\operatorname {sech} ^{2}\,x}
(
artanh
x
)
′
=
1
1
−
x
2
(
|
x
|
<
1
)
{\displaystyle (\operatorname {artanh} \,x)'={1 \over 1-x^{2}}(|x|<1)}
(
sech
x
)
′
=
−
tanh
x
sech
x
{\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}
(
arsech
x
)
′
=
−
1
x
1
−
x
2
(
0
<
x
<
1
)
{\displaystyle (\operatorname {arsech} \,x)'=-{1 \over x{\sqrt {1-x^{2}}}}(0 ( csch x ) ′ = − coth x csch x ( x ≠ 0 ) {\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x(x\neq 0)} ( arcsch x ) ′ = − 1 | x | 1 + x 2 ( x ≠ 0 ) {\displaystyle (\operatorname {arcsch} \,x)'=-{1 \over |x|{\sqrt {1+x^{2}}}}(x\neq 0)} ( coth x ) ′ = − csch 2 x ( x ≠ 0 ) {\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x(x\neq 0)} ( arcoth x ) ′ = 1 1 − x 2 ( | x | > 1 ) {\displaystyle (\operatorname {arcoth} \,x)'={1 \over 1-x^{2}}(|x|>1)} 特殊函数的导数 编辑 伽玛函数 d Γ ( x ) d x = ∫ 0 ∞ e − t t x − 1 ln t d t {\displaystyle {\frac {{\mbox{d}}\Gamma (x)}{{\mbox{d}}x}}=\int _{0}^{\infty }e^{-t}t^{x-1}\ln \!t{\mbox{d}}t} 註釋 编辑 ^ 這是前述廣義冪法則在「 f ( x ) = α {\displaystyle f(x)=\alpha } 且 g ( x ) = x {\displaystyle g(x)=x} 」時的特例。 ^ 這是前述廣義冪法則在「 f ( x ) = x {\displaystyle f(x)=x} 且 g ( x ) = x {\displaystyle g(x)=x} 」時的特例。