Derivative of xln(x)-x+1

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

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

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

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

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

The answer is:

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