Derivative of $$$\sin^{2}{\left(2 x \right)}$$$

The function $$$\sin^{2}{\left(2 x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = u^{2}$$$ and $$$g{\left(x \right)} = \sin{\left(2 x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = 2$$$:

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

Return to the old variable:

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

The function $$$\sin{\left(2 x \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ and $$$g{\left(x \right)} = 2 x$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

The derivative of the sine is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

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

Return to the old variable:

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

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = x$$$:

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

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

Simplify:

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

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