Derivative of y*exp^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 $1$

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

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

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

2. Now simplify:

   $$\left(y + 1\right) e^{y}$$

The answer is:

$$\left(y + 1\right) e^{y}$$