Derivative of cosx²

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. Now simplify:

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

The answer is:

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