Derivative of y*y*sin(y)^3

1. Apply the product rule:

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

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

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

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

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

   $h{\left(y \right)} = \sin^{3}{\left(y \right)}$; to find $\frac{d}{d y} h{\left(y \right)}$:

   1. Let $u = \sin{\left(y \right)}$.

   2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

      1. The derivative of sine is cosine:

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

      The result of the chain rule is:

      $$3 \sin^{2}{\left(y \right)} \cos{\left(y \right)}$$

   The result is: $3 y^{2} \sin^{2}{\left(y \right)} \cos{\left(y \right)} + 2 y \sin^{3}{\left(y \right)}$

2. Now simplify:

   $$y \left(3 y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin^{2}{\left(y \right)}$$

The answer is:

$$y \left(3 y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin^{2}{\left(y \right)}$$