Derivative of (3*log(x))/sqrt(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 $g{\left(x \right)} = \sqrt{x}$.

      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: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

      Now plug in to the quotient rule:

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

   So, the result is: $\frac{3 \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right)}{x}$

2. Now simplify:

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

The answer is: