Derivative of ycos(4y)

1. Apply the product rule:

   $$\frac{d}{d y} f{\left(y \right)} g{\left(y \right)} = f{\left(y \right)} \frac{d}{d y} g{\left(y \right)} + g{\left(y \right)} \frac{d}{d y} f{\left(y \right)}$$

   $f{\left(y \right)} = y$; to find $\frac{d}{d y} f{\left(y \right)}$:

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

   $g{\left(y \right)} = \cos{\left(4 y \right)}$; to find $\frac{d}{d y} g{\left(y \right)}$:

   1. Let $u = 4 y$.

   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 $\frac{d}{d y} 4 y$:

      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: $4$

      The result of the chain rule is:

      $$- 4 \sin{\left(4 y \right)}$$

   The result is: $- 4 y \sin{\left(4 y \right)} + \cos{\left(4 y \right)}$

The answer is:

$$- 4 y \sin{\left(4 y \right)} + \cos{\left(4 y \right)}$$