Derivative of y*log(y)-y

1. Differentiate $y \log{\left(y \right)} - y$ 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 $1$

      $g{\left(y \right)} = \log{\left(y \right)}$; 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: $\log{\left(y \right)} + 1$

   2. 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: $-1$

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

The answer is:

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