Derivative of cos^(2)4x

1. Let $u = \cos{\left(4 x \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(4 x \right)}$:

   1. Let $u = 4 x$.

   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 $\frac{d}{d x} 4 x$:

      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: $4$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$- 8 \sin{\left(4 x \right)} \cos{\left(4 x \right)}$$

The answer is:

$$- 8 \sin{\left(4 x \right)} \cos{\left(4 x \right)}$$