Derivative of sin(y)*cos(y)

1. Apply the product rule:

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

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

   1. The derivative of sine is cosine:

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

   $g{\left(y \right)} = \cos{\left(y \right)}$; 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: $- \sin^{2}{\left(y \right)} + \cos^{2}{\left(y \right)}$

2. Now simplify:

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

The answer is:

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