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脚本错误：没有“Message box”这个模块。
公式：
f
′
(
x
)
=
1
(
f
−
1
)
′
(
f
(
x
)
)
{\displaystyle {\color {CornflowerBlue}{f'}}(x)={\frac {1}{{\color {Salmon}{(f^{-1})'}}({\color {Blue}{f}}(x))}}}
x
0
≈
5.8
{\displaystyle x_{0}\approx 5.8}
f
′
(
x
0
)
=
1
4
{\displaystyle {\color {CornflowerBlue}{f'}}(x_{0})={\frac {1}{4}}}
(
f
−
1
)
′
(
f
(
x
0
)
)
=
4
{\displaystyle {\color {Salmon}{(f^{-1})'}}({\color {Blue}{f}}(x_{0}))=4~}
数学上，可導 雙射 函數
f
{\displaystyle f}
反函數微分 可由
f
{\displaystyle f}
f
′
{\displaystyle f'}
拉格朗日记法 ，反函数
f
−
1
{\displaystyle f^{-1}}
[註 1]
[
f
−
1
]
′
(
a
)
=
1
f
′
(
f
−
1
(
a
)
)
,
{\displaystyle \left[f^{-1}\right]'(a)={\frac {1}{f'\left(f^{-1}(a)\right)}},}
该表述等价于
D
[
f
−
1
]
=
1
(
D
f
)
∘
(
f
−
1
)
,
{\displaystyle {\mathcal {D}}\left[f^{-1}\right]={\frac {1}{({\mathcal {D}}f)\circ \left(f^{-1}\right)}},}
其中
D
{\displaystyle {\mathcal {D}}}
微分算子 （在函数的空间上），
∘
{\displaystyle \circ }
复合 算子。
記
y
=
f
(
x
)
{\displaystyle y=f(x)}
d
x
d
y
⋅
d
y
d
x
=
1.
{\displaystyle {\frac {dx}{dy}}\,\cdot \,{\frac {dy}{dx}}=1.}
換言之，函數及其反函數的导数均可逆[註 2] 链式规则 的直接结果，因为
d
x
d
y
⋅
d
y
d
x
=
d
x
d
x
,
{\displaystyle {\frac {dx}{dy}}\,\cdot \,{\frac {dy}{dx}}={\frac {dx}{dx}},}
而
x
{\displaystyle x}
x
{\displaystyle x}
几何上，函数和反函数有关于直线 y = x 斜率 变成其倒数 。
假设
f
{\displaystyle f}
x
{\displaystyle x}
反函数举例 [ 编辑 ]
y
=
x
2
{\displaystyle \,y=x^{2}}
x
{\displaystyle x}
x
=
y
{\displaystyle x={\sqrt {y}}}
d
y
d
x
=
2
x
;
d
x
d
y
=
1
2
y
=
1
2
x
{\displaystyle {\frac {dy}{dx}}=2x{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{2{\sqrt {y}}}}={\frac {1}{2x}}}
d
y
d
x
⋅
d
x
d
y
=
2
x
⋅
1
2
x
=
1.
{\displaystyle {\frac {dy}{dx}}\,\cdot \,{\frac {dx}{dy}}=2x\cdot {\frac {1}{2x}}=1.}
但是，在 x = 0
y
=
e
x
{\displaystyle \,y=e^{x}}
x
{\displaystyle x}
x
=
ln
y
{\displaystyle \,x=\ln {y}}
y
{\displaystyle y}
d
y
d
x
=
e
x
;
d
x
d
y
=
1
y
{\displaystyle {\frac {dy}{dx}}=e^{x}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{y}}}
d
y
d
x
⋅
d
x
d
y
=
e
x
⋅
1
y
=
e
x
e
x
=
1
{\displaystyle {\frac {dy}{dx}}\,\cdot \,{\frac {dx}{dy}}=e^{x}\cdot {\frac {1}{y}}={\frac {e^{x}}{e^{x}}}=1}
其他属性 [ 编辑 ]
f
−
1
(
x
)
=
∫
1
f
′
(
f
−
1
(
x
)
)
d
x
+
C
{\displaystyle {f^{-1}}(x)=\int {\frac {1}{f'({f^{-1}}(x))}}\,{dx}+\mathrm {C} }
[註 3] 可见，具有连续 导数的函数（光滑函数）在其导数非零的每一点的邻域 内都有反函数。如果导数不连续的，则上述积分公式不成立。
高阶导数 [ 编辑 ] 上面给出的链式法则 是通过对等式
x
=
f
−
1
(
f
(
x
)
)
{\displaystyle x=f^{-1}(f(x))}
x
{\displaystyle x}
x
{\displaystyle x}
d
2
y
d
x
2
⋅
d
x
d
y
+
d
d
x
(
d
x
d
y
)
⋅
(
d
y
d
x
)
=
0
,
{\displaystyle {\frac {d^{2}y}{dx^{2}}}\,\cdot \,{\frac {dx}{dy}}+{\frac {d}{dx}}\left({\frac {dx}{dy}}\right)\,\cdot \,\left({\frac {dy}{dx}}\right)=0,}
使用链式法则进一步简化为
d
2
y
d
x
2
⋅
d
x
d
y
+
d
2
x
d
y
2
⋅
(
d
y
d
x
)
2
=
0.
{\displaystyle {\frac {d^{2}y}{dx^{2}}}\,\cdot \,{\frac {dx}{dy}}+{\frac {d^{2}x}{dy^{2}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{2}=0.}
d
2
y
d
x
2
⋅
d
x
d
y
+
d
2
x
d
y
2
⋅
(
d
y
d
x
)
2
=
0
{\displaystyle {\frac {d^{2}y}{dx^{2}}}\,\cdot \,{\frac {dx}{dy}}+{\frac {d^{2}x}{dy^{2}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{2}=0}
用之前得到的恒等式替换一阶导数，得到
d
2
y
d
x
2
=
−
d
2
x
d
y
2
⋅
(
d
y
d
x
)
3
{\displaystyle {\frac {d^{2}y}{dx^{2}}}=-{\frac {d^{2}x}{dy^{2}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{3}}
d
2
y
d
x
2
=
−
d
2
x
d
y
2
⋅
(
d
y
d
x
)
3
.
{\displaystyle {\frac {d^{2}y}{dx^{2}}}=-{\frac {d^{2}x}{dy^{2}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{3}.}
对三阶导数类似：
d
3
y
d
x
3
=
−
d
3
x
d
y
3
⋅
(
d
y
d
x
)
4
−
3
d
2
x
d
y
2
⋅
d
2
y
d
x
2
⋅
(
d
y
d
x
)
2
{\displaystyle {\frac {d^{3}y}{dx^{3}}}=-{\frac {d^{3}x}{dy^{3}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{4}-3{\frac {d^{2}x}{dy^{2}}}\,\cdot \,{\frac {d^{2}y}{dx^{2}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{2}}
或者用二阶导数 的公式，
d
3
y
d
x
3
=
−
d
3
x
d
y
3
⋅
(
d
y
d
x
)
4
+
3
(
d
2
x
d
y
2
)
2
⋅
(
d
y
d
x
)
5
{\displaystyle {\frac {d^{3}y}{dx^{3}}}=-{\frac {d^{3}x}{dy^{3}}}\,\cdot \,\left({\frac {dy}{dx}}\right)^{4}+3\left({\frac {d^{2}x}{dy^{2}}}\right)^{2}\,\cdot \,\left({\frac {dy}{dx}}\right)^{5}}
这些公式是由Faa di Bruno公式推广。
这些公式也可以用拉格朗日表示法来表示。如果
f
{\displaystyle f}
g
{\displaystyle g}
g
″
(
x
)
=
−
f
″
(
g
(
x
)
)
[
f
′
(
g
(
x
)
)
]
3
{\displaystyle g''(x)={\frac {-f''(g(x))}{[f'(g(x))]^{3}}}}
反函数的微分举例 [ 编辑 ]
y
=
e
x
{\displaystyle \,y=e^{x}}
x
=
ln
y
{\displaystyle \,x=\ln y}
d
y
d
x
=
d
2
y
d
x
2
=
e
x
=
y
;
(
d
y
d
x
)
3
=
y
3
;
{\displaystyle {\frac {dy}{dx}}={\frac {d^{2}y}{dx^{2}}}=e^{x}=y{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}\left({\frac {dy}{dx}}\right)^{3}=y^{3};}
于是，
d
2
x
d
y
2
⋅
y
3
+
y
=
0
;
d
2
x
d
y
2
=
−
1
y
2
{\displaystyle {\frac {d^{2}x}{dy^{2}}}\,\cdot \,y^{3}+y=0{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {d^{2}x}{dy^{2}}}=-{\frac {1}{y^{2}}}}
与直接计算相同。
REDIRECT Template:Delete 脚本错误：没有“Redirect_Template_List”这个模块。 This article "反函数的微分" is from Wikipedia . The list of its authors can be seen in its historical and/or the page Edithistory:反函数的微分 . Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
This article "反函数的微分" is from Wikipedia . The list of its authors can be seen in its historical and/or the page Edithistory:反函数的微分 . Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
This article "反函数的微分" is from Wikipedia . The list of its authors can be seen in its historical and/or the page Edithistory:反函数的微分 . Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
![{\displaystyle \left[f^{-1}\right]'(a)={\frac {1}{f'\left(f^{-1}(a)\right)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0347bdeee0d3f1ccf902a27310a5e388d8665c73)
![{\displaystyle {\mathcal {D}}\left[f^{-1}\right]={\frac {1}{({\mathcal {D}}f)\circ \left(f^{-1}\right)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b71cf74aa5480ff9d43a9823ceb5580914f923a5)
![{\displaystyle g''(x)={\frac {-f''(g(x))}{[f'(g(x))]^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c374bf89ec0cde43e6ec3e86cbf86490e55f35d)
