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

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

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

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

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

      The result is: $1$

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

      1. Apply the quotient rule, which is:

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

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

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

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

         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:

         $\frac{1 - \log{\left(x \right)}}{x^{2}}$

      So, the result is: $- \frac{1 - \log{\left(x \right)}}{x^{2}}$

   The result is: $1 - \frac{1 - \log{\left(x \right)}}{x^{2}}$

2. Now simplify:

   $$\frac{x^{2} + \log{\left(x \right)} - 1}{x^{2}}$$

The answer is:

$$\frac{x^{2} + \log{\left(x \right)} - 1}{x^{2}}$$