Derivative of f(x)=xsin²(2x)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x$ goes to $1$

   $g{\left(x \right)} = \sin^{2}{\left(2 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \sin{\left(2 x \right)}$.

   2. Apply the power rule: $u^{2}$ goes to $2 u$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x \right)}$:

      1. Let $u = 2 x$.

      2. The derivative of sine is cosine:

         $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $2$

         The result of the chain rule is:

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

      The result of the chain rule is:

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

   The result is: $4 x \sin{\left(2 x \right)} \cos{\left(2 x \right)} + \sin^{2}{\left(2 x \right)}$

2. Now simplify:

   $$2 x \sin{\left(4 x \right)} - \frac{\cos{\left(4 x \right)}}{2} + \frac{1}{2}$$

The answer is:

$$2 x \sin{\left(4 x \right)} - \frac{\cos{\left(4 x \right)}}{2} + \frac{1}{2}$$