Derivative of (2lnx)/x

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:

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

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

The answer is:

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