Derivative of cos(2x)^(5)

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

2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

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

   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:

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

   The result of the chain rule is:

   $$- 10 \sin{\left(2 x \right)} \cos^{4}{\left(2 x \right)}$$

The answer is:

$$- 10 \sin{\left(2 x \right)} \cos^{4}{\left(2 x \right)}$$