求y=√[[(x-1)(x-2)]/[(x-3)(x-4)]]的导数

oyhk 学习笔记

例6:求 y = \sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)}}的导数。

解:先在等式两边取对数(假定 x > 4),得
y=\left[ \dfrac{(x-1)(x-2)}{(x-3)(x-4)} \right]^\dfrac{1}{2}
\ln y = \ln \left[ \dfrac{(x-1)(x-2)}{(x-3)(x-4)} \right]^\dfrac{1}{2}

\ln y = \dfrac{1}{2} \left[\ln \dfrac{(x-1)(x-2)}{(x-3)(x-4)} \right]

\ln y = \dfrac{1}{2} \left[ \ln(x-1)+ln(x-2)-ln(x-3)-ln(x-4)\right]

上式两边对x求导,注意到y=y(x),得

\left(\ln y\right)^\prime = \left[\dfrac{1}{2} \left[ \ln(x-1)+ln(x-2)-ln(x-3)-ln(x-4)\right] \right]^\prime

\dfrac{1}{y}y^\prime = \dfrac{1}{2}\left[\dfrac{1}{x-1}+\dfrac{1}{x-2}-\dfrac{1}{x-3}-\dfrac{1}{x-4}\right]

于是

y^\prime=\dfrac{y}{2}\left[\dfrac{1}{x-1}+\dfrac{1}{x-2}-\dfrac{1}{x-3}-\dfrac{1}{x-4}\right]

同理求出:

当 x < 1 时,y = \sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)}}
当 2 < x < 3 时,y = \sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)}}

这里要记得对数函数的运算公式:

\log_ab + \log_ac = \log_abc

\log_ab – \log_ac = \log_a\dfrac{b}{c}