$$$\ln\left(\sin{\left(x \right)}\right)$$$の導関数

関数$$$\ln\left(\sin{\left(x \right)}\right)$$$は、2つの関数$$$f{\left(u \right)} = \ln\left(u\right)$$$と$$$g{\left(x \right)} = \sin{\left(x \right)}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

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

自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

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

元の変数に戻す:

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

正弦関数の導関数は$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

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

簡単化せよ:

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

したがって、$$$\frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right) = \frac{1}{\tan{\left(x \right)}}$$$。