Derivative of (ye^y)+y^2

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

   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 $1$

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

      1. The derivative of $e^{y}$ is itself.

      The result is: $y e^{y} + e^{y}$

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

   The result is: $y e^{y} + 2 y + e^{y}$

2. Now simplify:

   $$y e^{y} + 2 y + e^{y}$$

The answer is:

$$y e^{y} + 2 y + e^{y}$$