Derivative of 4cos^5x-sqrt(lnx)

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

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

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

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

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

         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:

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

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

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

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

      2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

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

         1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

         The result of the chain rule is:

         $$\frac{1}{2 x \sqrt{\log{\left(x \right)}}}$$

      So, the result is: $- \frac{1}{2 x \sqrt{\log{\left(x \right)}}}$

   The result is: $- 20 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - \frac{1}{2 x \sqrt{\log{\left(x \right)}}}$

The answer is:

$$- 20 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - \frac{1}{2 x \sqrt{\log{\left(x \right)}}}$$