Derivative of y*lny

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 $1$

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

   1. The derivative of $\log{\left(y \right)}$ is $\frac{1}{y}$.

   The result is: $\log{\left(y \right)} + 1$

The answer is:

$$\log{\left(y \right)} + 1$$