Derivative of 4^cos(5x)

1. Let $u = \cos{\left(5 x \right)}$.

2. $\frac{d}{d u} 4^{u} = 4^{u} \log{\left(4 \right)}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x \right)}$:

   1. Let $u = 5 x$.

   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 $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$- 5 \sin{\left(5 x \right)}$$

   The result of the chain rule is:

   $$- 5 \cdot 4^{\cos{\left(5 x \right)}} \log{\left(4 \right)} \sin{\left(5 x \right)}$$

4. Now simplify:

   $$- 10 \cdot 4^{\cos{\left(5 x \right)}} \log{\left(2 \right)} \sin{\left(5 x \right)}$$

The answer is:

$$- 10 \cdot 4^{\cos{\left(5 x \right)}} \log{\left(2 \right)} \sin{\left(5 x \right)}$$