Derivative of xlogx-(x-1)

1. Differentiate $x \log{\left(x \right)} + \left(1 - x\right)$ term by term:

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

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

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

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

   2. Differentiate $1 - x$ term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $-1$

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

      The result is: $-1$

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

The answer is:

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