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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., \( \sqrt{q} = q^{1/2} \)).

Find the derivative of p(q) = -π/3 * √q^3.

Learn how to find the derivative of the function p(q) = -π/3 * √q^3 using differentiation rules. This step-by-step solution simplifies the expression and is suitable for high school calculus students.

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