Derivative of 1/x+2ln(x)-(ln(x)/x)

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

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

      1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$

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

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

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

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

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

2. Now simplify:

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

The answer is:

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