The derivative of the given function $p(q) = -\frac{\pi}{3} \sqrt{q^3}$ is:

$p'(q) = -\frac{\pi \sqrt{q^3}}{2q}$

This expression simplifies to:

$p'(q) = -\frac{\pi}{2} \sqrt{q}$

Thus, the correct answer is:

$p'(q) = -\frac{\pi}{2} \sqrt{q}$

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the derivative of a power of $q$?
2. What rule is used when differentiating square roots?
3. What is the derivative of $\sqrt{q^3}$ in detail?
4. How do constants like $\pi$ affect differentiation?
5. How does simplification affect the final derivative expression?

Tip: When differentiating a square root, rewrite it as an exponent to make it easier (i.e., $\sqrt{q} = q^{1/2}$).