Derivative of y-cos(2*y)

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

   1. Apply the power rule: $y$ goes to $1$

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

      1. Let $u = 2 y$.

      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 $\frac{d}{d y} 2 y$:

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

            1. Apply the power rule: $y$ goes to $1$

            So, the result is: $2$

         The result of the chain rule is:

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

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

   The result is: $2 \sin{\left(2 y \right)} + 1$

The answer is:

$$2 \sin{\left(2 y \right)} + 1$$