$$$\ln\left(\cos{\left(x \right)}\right)$$$的导数

函数$$$\ln\left(\cos{\left(x \right)}\right)$$$是两个函数$$$f{\left(u \right)} = \ln\left(u\right)$$$和$$$g{\left(x \right)} = \cos{\left(x \right)}$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。

应用链式法则 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$

自然对数的导数为 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$

返回到原变量：

$$\frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(\cos{\left(x \right)}\right)}}$$

余弦函数的导数是$$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$：

$$\frac{{\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}}{\cos{\left(x \right)}} = \frac{{\color{red}\left(- \sin{\left(x \right)}\right)}}{\cos{\left(x \right)}}$$

化简：

$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} = - \tan{\left(x \right)}$$

因此，$$$\frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right) = - \tan{\left(x \right)}$$$。