Derivative of y*cos(y)

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 $1$

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

   1. The derivative of cosine is negative sine:

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

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

The answer is:

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