Derivative of 3cos^2x

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: $u^{2}$ 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:

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

   So, the result is: $- 6 \sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

   $$- 3 \sin{\left(2 x \right)}$$

The answer is: