Derivative of 2cos(y)^2

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

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

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

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

      1. The derivative of cosine is negative sine:

         $$\frac{d}{d y} \cos{\left(y \right)} = - \sin{\left(y \right)}$$

      The result of the chain rule is:

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

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

2. Now simplify:

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

The answer is:

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