Derivative of log(x)^2/sqrt(x)

1. Apply the quotient rule, which is:

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

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

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

   1. Let $u = \log{\left(x \right)}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$:

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

      The result of the chain rule is:

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

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

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

   Now plug in to the quotient rule:

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

2. Now simplify:

   $$\frac{\left(4 - \log{\left(x \right)}\right) \log{\left(x \right)}}{2 x^{\frac{3}{2}}}$$

The answer is: