Derivative of lnx-2(x-1)/x

1. Differentiate $\log{\left(x \right)} - \frac{2 \left(x - 1\right)}{x}$ term by term:

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

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

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

         1. Apply the quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = x - 1$ and $g{\left(x \right)} = x$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

               The result is: $1$

            To find $\frac{d}{d x} g{\left(x \right)}$:

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

            Now plug in to the quotient rule:

            $\frac{1}{x^{2}}$

         So, the result is: $\frac{2}{x^{2}}$

      So, the result is: $- \frac{2}{x^{2}}$

   The result is: $\frac{1}{x} - \frac{2}{x^{2}}$

2. Now simplify:

   $$\frac{x - 2}{x^{2}}$$

The answer is:

$$\frac{x - 2}{x^{2}}$$