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目录
移至侧栏
隐藏
序言
1
运算法则
2
幂函数和多项式
3
三角函数
4
指数和对数函数
5
反三角函数
6
双曲和反双曲函数
开关目录
微积分学/导数表
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维基教科书,自由的教学读本
<
微积分学
← 求和符号
微积分学
积分表 →
导数表
运算法则
[
编辑
]
d
d
x
(
f
+
g
)
=
d
f
d
x
+
d
g
d
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(f+g)={\frac {\mathrm {d} f}{\mathrm {d} x}}+{\frac {\mathrm {d} g}{\mathrm {d} x}}}
d
d
x
(
c
⋅
f
)
=
c
⋅
d
f
d
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(c\cdot f)=c\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}}
d
d
x
(
f
⋅
g
)
=
f
⋅
d
g
d
x
+
g
⋅
d
f
d
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(f\cdot g)=f\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}+g\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}}
d
d
x
(
f
g
)
=
−
f
⋅
d
g
d
x
+
g
⋅
d
f
d
x
g
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {f}{g}}\right)={\dfrac {-f\cdot {\dfrac {\mathrm {d} g}{dx}}+g\cdot {\dfrac {\mathrm {d} f}{\mathrm {d} x}}}{g^{2}}}}
d
d
x
[
f
(
g
(
x
)
)
]
=
d
f
d
g
⋅
d
g
d
x
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}[f(g(x))]={\frac {\mathrm {d} f}{\mathrm {d} g}}\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}=f'(g(x))\cdot g'(x)}
d
n
d
x
n
f
(
x
)
g
(
x
)
=
∑
i
=
0
n
(
n
i
)
f
(
n
−
i
)
(
x
)
g
(
i
)
(
x
)
{\displaystyle {\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}f(x)g(x)=\sum _{i=0}^{n}\left({\begin{matrix}n\\i\end{matrix}}\right)f^{(n-i)}(x)g^{(i)}(x)}
d
d
x
(
1
f
)
=
−
f
′
f
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {1}{f}}\right)=-{\frac {f'}{f^{2}}}}
幂函数和多项式
[
编辑
]
d
d
x
(
c
)
=
0
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(c)=0}
d
d
x
x
=
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}x=1}
d
d
x
x
n
=
n
x
n
−
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}x^{n}=nx^{n-1}}
d
d
x
x
=
1
2
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}}
d
d
x
1
x
=
−
1
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}}
d
d
x
(
c
n
x
n
+
c
n
−
1
x
n
−
1
+
c
n
−
2
x
n
−
2
+
⋯
+
c
2
x
2
+
c
1
x
+
c
0
)
=
n
c
n
x
n
−
1
+
(
n
−
1
)
c
n
−
1
x
n
−
2
+
(
n
−
2
)
c
n
−
2
x
n
−
3
+
⋯
+
2
c
2
x
+
c
1
{\displaystyle {{\frac {\mathrm {d} }{\mathrm {d} x}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}}}
三角函数
[
编辑
]
d
d
x
sin
(
x
)
=
cos
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sin(x)=\cos(x)}
d
d
x
cos
(
x
)
=
−
sin
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cos(x)=-\sin(x)}
d
d
x
tan
(
x
)
=
sec
2
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\tan(x)=\sec ^{2}(x)}
d
d
x
cot
(
x
)
=
−
csc
2
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cot(x)=-\csc ^{2}(x)}
d
d
x
sec
(
x
)
=
sec
(
x
)
tan
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sec(x)=\sec(x)\tan(x)}
d
d
x
csc
(
x
)
=
−
csc
(
x
)
cot
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\csc(x)=-\csc(x)\cot(x)}
指数和对数函数
[
编辑
]
d
d
x
e
x
=
e
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}}
d
d
x
a
x
=
a
x
ln
(
a
)
a
>
0
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}a^{x}=a^{x}\ln(a)\qquad a>0}
d
d
x
ln
(
x
)
=
1
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}}
d
d
x
log
a
(
x
)
=
1
x
ln
(
a
)
a
>
0
,
a
≠
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\log _{a}(x)={\frac {1}{x\ln(a)}}\qquad a>0\ ,\ a\neq 1}
d
d
x
(
f
g
)
=
d
d
x
(
e
g
ln
(
f
)
)
=
f
g
(
f
′
g
f
+
g
′
ln
(
f
)
)
f
>
0
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(f^{g})={\frac {\mathrm {d} }{\mathrm {d} x}}\left(e^{g\ln(f)}\right)=f^{g}\left(f'{\frac {g}{f}}+g'\ln(f)\right)\qquad f>0}
d
d
x
(
c
f
)
=
d
d
x
(
e
f
ln
(
c
)
)
=
c
f
ln
(
c
)
⋅
f
′
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(c^{f})={\frac {\mathrm {d} }{\mathrm {d} x}}\left(e^{f\ln(c)}\right)=c^{f}\ln(c)\cdot f'}
反三角函数
[
编辑
]
d
d
x
arcsin
(
x
)
=
1
1
−
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}}
d
d
x
arccos
(
x
)
=
−
1
1
−
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}}
d
d
x
arctan
(
x
)
=
1
x
2
+
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arctan(x)={\frac {1}{x^{2}+1}}}
d
d
x
arccot
(
x
)
=
−
1
x
2
+
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccot}(x)=-{\frac {1}{x^{2}+1}}}
d
d
x
arcsec
(
x
)
=
1
|
x
|
x
2
−
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsec}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d
d
x
arccsc
(
x
)
=
−
1
|
x
|
x
2
−
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccsc}(x)=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
双曲和反双曲函数
[
编辑
]
d
d
x
sinh
(
x
)
=
cosh
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sinh(x)=\cosh(x)}
d
d
x
cosh
(
x
)
=
sinh
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cosh(x)=\sinh(x)}
d
d
x
tanh
(
x
)
=
s
e
c
h
2
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\tanh(x)={\rm {sech}}^{2}(x)}
d
d
x
s
e
c
h
(
x
)
=
−
tanh
(
x
)
s
e
c
h
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {sech}}(x)=-\tanh(x){\rm {sech}}(x)}
d
d
x
coth
(
x
)
=
−
c
s
c
h
2
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\coth(x)=-{\rm {csch}}^{2}(x)}
d
d
x
c
s
c
h
(
x
)
=
−
coth
(
x
)
c
s
c
h
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {csch}}(x)=-\coth(x){\rm {csch}}(x)}
d
d
x
a
r
s
i
n
h
(
x
)
=
1
1
+
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsinh}}(x)={\frac {1}{\sqrt {1+x^{2}}}}}
d
d
x
a
r
c
o
s
h
(
x
)
=
1
x
2
−
1
,
x
>
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcosh}}(x)={\frac {1}{\sqrt {x^{2}-1}}}\ ,\ x>1}
d
d
x
a
r
t
a
n
h
(
x
)
=
1
1
−
x
2
,
|
x
|
<
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {artanh}}(x)={\frac {1}{1-x^{2}}}\ ,\ |x|<1}
d
d
x
a
r
c
s
c
h
(
x
)
=
−
1
|
x
|
1
+
x
2
,
x
≠
0
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcsch}}(x)=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}\ ,\ x\neq 0}
d
d
x
a
r
s
e
c
h
(
x
)
=
−
1
x
1
−
x
2
,
0
<
x
<
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsech}}(x)=-{\frac {1}{x{\sqrt {1-x^{2}}}}}\ ,\ 0<x<1}
d
d
x
a
r
c
o
t
h
(
x
)
=
1
1
−
x
2
,
|
x
|
>
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcoth}}(x)={\frac {1}{1-x^{2}}}\ ,\ |x|>1}
← 求和符号
微积分学
积分表 →
导数表
章节导航
:
目录
·
预备知识
·
极限
·
导数
·
积分
·
极坐标方程与参数方程
·
数列和级数
·
多元函数微积分
·
扩展知识
·
附录
分类
:
微积分学