Derivative of x/lnx

1. Apply the quotient rule, which is:

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

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

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

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

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

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

   Now plug in to the quotient rule:

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

The answer is:

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