Derivative of 0,5-cos(cos(y)+2)

1. Differentiate $\frac{1}{2} - \cos{\left(\cos{\left(y \right)} + 2 \right)}$ term by term:

   1. The derivative of the constant $\frac{1}{2}$ is zero.

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

      1. Let $u = \cos{\left(y \right)} + 2$.

      2. The derivative of cosine is negative sine:

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \left(\cos{\left(y \right)} + 2\right)$:

         1. Differentiate $\cos{\left(y \right)} + 2$ term by term:

            1. The derivative of cosine is negative sine:

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

            2. The derivative of the constant $2$ is zero.

            The result is: $- \sin{\left(y \right)}$

         The result of the chain rule is:

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

      So, the result is: $- \sin{\left(y \right)} \sin{\left(\cos{\left(y \right)} + 2 \right)}$

   The result is: $- \sin{\left(y \right)} \sin{\left(\cos{\left(y \right)} + 2 \right)}$

2. Now simplify:

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

The answer is:

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