反函数求导

求函数 y = tan ⁡ a x y=\tan ax y=tanax的反函数 x = x ( y ) x=x(y) x=x(y)的导数。

解： ∵ \qquad \because ∵函数 y = tan ⁡ a x y=\tan ax y=tanax严格单调递增且可导

\qquad 且 y ′ = a cos ⁡ 2 a x y'=\dfrac{a}{\cos^2 ax} y′=cos2axa​

∴ x = x ( y ) \qquad \therefore x=x(y) ∴x=x(y)可导

x ′ ( y ) = 1 y ′ ( x ) = c o s 2 a x a \qquad x'(y)=\dfrac{1}{y'(x)}=\dfrac{cos^2ax}{a} x′(y)=y′(x)1​=acos2ax​

∵ cos ⁡ 2 a x ( 1 + tan ⁡ 2 a x ) = cos ⁡ 2 a x + sin ⁡ 2 a x = 1 \qquad \because \cos^2ax(1+\tan^2 ax)=\cos^2ax+\sin^2 ax=1 ∵cos2ax(1+tan2ax)=cos2ax+sin2ax=1

∴ cos ⁡ 2 a x = 1 1 + tan ⁡ 2 a x \qquad \therefore \cos^2ax=\dfrac{1}{1+\tan^2 ax} ∴cos2ax=1+tan2ax1​

∴ x ′ ( y ) = cos ⁡ 2 a x a = 1 a + a tan ⁡ 2 a x = 1 a + a y 2 \qquad \therefore x'(y)=\dfrac{\cos^2ax}{a}=\dfrac{1}{a+a\tan^2 ax}=\dfrac{1}{a+ay^2} ∴x′(y)=acos2ax​=a+atan2ax1​=a+ay21​

求函数 y = x + e x y=x+e^x y=x+ex的反函数 x = x ( y ) x=x(y) x=x(y)的导数。

解： ∵ \qquad \because ∵函数 y = x + e x y=x+e^x y=x+ex严格单调递增且可导

\qquad 且 y ′ = 1 + e x y'=1+e^x y′=1+ex

∴ x ′ ( y ) = 1 y ′ ( x ) = 1 1 + e x \qquad \therefore x'(y)=\dfrac{1}{y'(x)}=\dfrac{1}{1+e^x} ∴x′(y)=y′(x)1​=1+ex1​