Derivative of ((1\4)sin²(2x))

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

   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:

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

   So, the result is: $\sin{\left(2 x \right)} \cos{\left(2 x \right)}$

2. Now simplify:

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

The answer is:

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