Derivative of 1/3sqrt(x^5)

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

   1. Let $u = x^{5}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{5}$:

      1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

      The result of the chain rule is:

      $$\frac{5 x^{4}}{2 \sqrt{x^{5}}}$$

   So, the result is: $\frac{5 x^{4}}{6 \sqrt{x^{5}}}$

The answer is:

$$\frac{5 x^{4}}{6 \sqrt{x^{5}}}$$