Derivative of -y(y^2+1)

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. 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: $-1$

   $g{\left(y \right)} = y^{2} + 1$; to find $\frac{d}{d y} g{\left(y \right)}$:

   1. Differentiate $y^{2} + 1$ term by term:

      1. Apply the power rule: $y^{2}$ goes to $2 y$

      2. The derivative of the constant $1$ is zero.

      The result is: $2 y$

   The result is: $- 3 y^{2} - 1$

The answer is:

$$- 3 y^{2} - 1$$