Derivative of log(√(12x))^⅓

1. Let $u = \log{\left(\sqrt{12 x} \right)}$.

2. Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\sqrt{12 x} \right)}$:

   1. Let $u = \sqrt{12 x}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{12 x}$:

      1. Let $u = 12 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 12 x$:

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

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

            So, the result is: $12$

         The result of the chain rule is:

         $$\frac{\sqrt{3}}{\sqrt{x}}$$

      The result of the chain rule is:

      $$\frac{1}{2 x}$$

   The result of the chain rule is:

   $$\frac{1}{6 x \log{\left(\sqrt{12 x} \right)}^{\frac{2}{3}}}$$

4. Now simplify:

   $$\frac{1}{6 x \left(\frac{\log{\left(x \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + \log{\left(2 \right)}\right)^{\frac{2}{3}}}$$

The answer is:

$$\frac{1}{6 x \left(\frac{\log{\left(x \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + \log{\left(2 \right)}\right)^{\frac{2}{3}}}$$