Derivative of 4sin2xcos2x

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

   1. Apply the product rule:

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

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

      1. Let $u = 2 x$.

      2. The derivative of cosine is negative sine:

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

      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:

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

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

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

   So, the result is: $- 8 \sin^{2}{\left(2 x \right)} + 8 \cos^{2}{\left(2 x \right)}$

The answer is:

$$- 8 \sin^{2}{\left(2 x \right)} + 8 \cos^{2}{\left(2 x \right)}$$