Apply the product rule $$$ \frac{d}{dx} \left(f{\left(x \ri

in other words， $$$ \frac{d}{dx} \left(x\right) = 1 $$$ : $$ 2 x \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \sin{\left(2 x \right)} = 2 x \cos{\left(2 x \right)} {\color{red}\left(1\right)} + \sin{\left(2 x \right)} $$ Thus， in other words， Apply the product rule $$$ \frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right) $$$ with $$$ f{\left(x \right)} = x $$$ and $$$ g{\left(x \right)} = \sin{\left(2 x \right)} $$$ : $$ {\color{red}\left(\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(2 x \right)} + x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)} $$ Apply the power rule $$$ \frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} $$$ with $$$ n = 1 $$$ ， $$$ \frac{d}{dx} \left(x\right) = 1 $$$ : $$ x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right) + \sin{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right) + \sin{\left(2 x \right)} {\color{red}\left(1\right)} $$ 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 $$$ \frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right) $$$ : $$ x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)} + \sin{\left(2 x \right)} = x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)} + \sin{\left(2 x \right)} $$ The derivative of the sine is $$$ \frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)} $$$ : $$ x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} = x {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} $$ Return to the old variable: $$ x \cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} = x \cos{\left({\color{red}\left(2 x\right)} \right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} $$ 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 $$$ f{\left(x \right)} = x $$$ : $$ x \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} + \sin{\left(2 x \right)} = x \cos{\left(2 x \right)} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)} + \sin{\left(2 x \right)} $$ Apply the power rule $$$ \frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} $$$ with $$$ n = 1 $$$ ， $$$ \frac{d}{dx} \left(x \sin{\left(2 x \right)}\right) = 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)} $$$ . ，。