Derivative of (ln(x)-1)x

1. Apply the product rule:

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

   $f{\left(x \right)} = \log{\left(x \right)} - 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

      The result is: $\frac{1}{x}$

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

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

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

The answer is:

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