$$$\sin^{2}{\left(x \right)}$$$の導関数

関数$$$\sin^{2}{\left(x \right)}$$$は、2つの関数$$$f{\left(u \right)} = u^{2}$$$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:

$${\color{red}\left(\frac{d}{dx} \left(\sin^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$

冪法則 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ を $$$n = 2$$$ に対して適用する:

$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$

元の変数に戻す:

$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = 2 {\color{red}\left(\sin{\left(x \right)}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$

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

$$2 \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = 2 \sin{\left(x \right)} {\color{red}\left(\cos{\left(x \right)}\right)}$$

簡単化せよ:

$$2 \sin{\left(x \right)} \cos{\left(x \right)} = \sin{\left(2 x \right)}$$

したがって、$$$\frac{d}{dx} \left(\sin^{2}{\left(x \right)}\right) = \sin{\left(2 x \right)}$$$。