$$$\ln\left(\sin{\left(x \right)}\right)$$$ 的導數

函數 $$$\ln\left(\sin{\left(x \right)}\right)$$$ 是兩個函數 $$$f{\left(u \right)} = \ln\left(u\right)$$$ 與 $$$g{\left(x \right)} = \sin{\left(x \right)}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。

應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：

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

自然對數的導數為 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：

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

返回原變數：

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

正弦函數的導數為$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$：

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

化簡：

$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = \frac{1}{\tan{\left(x \right)}}$$

因此，$$$\frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right) = \frac{1}{\tan{\left(x \right)}}$$$。