Derivative of 2sin^2xcos^2x

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

   1. Apply the product rule:

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

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

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

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

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

         1. The derivative of cosine is negative sine:

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

         The result of the chain rule is:

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

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

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

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

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

         1. The derivative of sine is cosine:

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

         The result of the chain rule is:

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

      The result is: $- 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$

   So, the result is: $- 4 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$

2. Now simplify:

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

The answer is:

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