$$$\sin^{2}{\left(2 x \right)}$$$的导数

函数$$$\sin^{2}{\left(2 x \right)}$$$是两个函数$$$f{\left(u \right)} = u^{2}$$$和$$$g{\left(x \right)} = \sin{\left(2 x \right)}$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。

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

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

应用幂次法则 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$，其中 $$$n = 2$$$:

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

返回到原变量：

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

函数$$$\sin{\left(2 x \right)}$$$是两个函数$$$f{\left(u \right)} = \sin{\left(u \right)}$$$和$$$g{\left(x \right)} = 2 x$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。

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

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

正弦函数的导数为 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

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

返回到原变量：

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

对 $$$c = 2$$$ 和 $$$f{\left(x \right)} = x$$$ 应用常数倍法则 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$：

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

应用幂法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是说，$$$\frac{d}{dx} \left(x\right) = 1$$$：

$$4 \sin{\left(2 x \right)} \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 4 \sin{\left(2 x \right)} \cos{\left(2 x \right)} {\color{red}\left(1\right)}$$

化简：

$$4 \sin{\left(2 x \right)} \cos{\left(2 x \right)} = 2 \sin{\left(4 x \right)}$$

因此，$$$\frac{d}{dx} \left(\sin^{2}{\left(2 x \right)}\right) = 2 \sin{\left(4 x \right)}$$$。