Derivative of cos^2x-3x^5

1. Differentiate $- 3 x^{5} + \cos^{2}{\left(x \right)}$ term by term:

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

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

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

      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:

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

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

      1. 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: $15 x^{4}$

      So, the result is: $- 15 x^{4}$

   The result is: $- 15 x^{4} - 2 \sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

   $$- 15 x^{4} - \sin{\left(2 x \right)}$$

The answer is:

$$- 15 x^{4} - \sin{\left(2 x \right)}$$