Derivative of sin(cosx)+4x^5

1. Differentiate $4 x^{5} + \sin{\left(\cos{\left(x \right)} \right)}$ term by term:

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

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

      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:

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

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

      1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

      So, the result is: $20 x^{4}$

   The result is: $20 x^{4} - \sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} \right)}$

The answer is:

$$20 x^{4} - \sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} \right)}$$