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如果函数\(x = f(y)\)在区间\(I_y\)内单调、可导且\(f′(y)≠0\),那么它的反函数\(y…

Inverse Derivative Formula

2026-07-06 15:56
2026-07-06 20:11

如果函数\(x = f(y)\)在区间\(I_y\)内单调、可导且\(f′(y)≠0\),那么它的反函数\(y = f^{−1}(x)\)在区间\(I_x=\{x|x = f(y),y∈I_y\}\)内也可导,且\((f^{−1}(x))′ = \frac{1}{f′(y)}\)或\(\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\),即反函数的导数等于直接函数导数的倒数。这是反函数求导的基本公式。另外,对于反导数,若\(y = f\),则其反导数为\(x = φ\),这是反导数的一种基本表示形式。在求反函数导数时,还可利用链式法则,具体为:将原函数与反函数看作是一个复合函数,首先求出原函数的导数,然后将原函数的导数与反函数的导数相乘,最后将结果除以原函数的值(如\((\arcsin x)′=\frac{1}{x}\)或\(\frac{1}{\sin y}\))。 点击前往免费阅读更多精彩小说

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