Derivative of ln(x)/(sqrt(x+2*x))

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = \sqrt{3} \sqrt{x}$.

   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. The derivative of a constant times a function is the constant times the derivative of the function.

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

      So, the result is: $\frac{\sqrt{3}}{2 \sqrt{x}}$

   Now plug in to the quotient rule:

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

2. Now simplify:

   $$\frac{\sqrt{3} \left(2 - \log{\left(x \right)}\right)}{6 x^{\frac{3}{2}}}$$

The answer is: