Derivative of (x+lnx)/x^2

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = x + \log{\left(x \right)}$ and $g{\left(x \right)} = x^{2}$.

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

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

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

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

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

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

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

   Now plug in to the quotient rule:

   $\frac{x^{2} \left(1 + \frac{1}{x}\right) - 2 x \left(x + \log{\left(x \right)}\right)}{x^{4}}$

2. Now simplify:

   $$\frac{- x - 2 \log{\left(x \right)} + 1}{x^{3}}$$

The answer is:

$$\frac{- x - 2 \log{\left(x \right)} + 1}{x^{3}}$$