If the volume of a frustrum is given as $V = \frac{\pi h}{3} (R^2 + Rr + r^2)$, find $\frac{\mathrm d V}{\mathrm d R}$.

Analyze the question

We need the derivative of V with respect to R. This means R is the variable, while h and r are treated as constants.

Concept

Use the power rule, d/dR (Rⁿ) = nRⁿ⁻¹, and the constant multiple rule. Any term without R differentiates to 0.

Solution

V = (πh/3)(R² + Rr + r²)

• d/dR (R²) = 2R
• d/dR (Rr) = r, since r is a constant
• d/dR (r²) = 0, since r² is a constant

Therefore:

dV/dR = (πh/3)(2R + r)

Checking the options

• A: (πh/3)(2R + r). This matches our result.
• B: The factor πh/3 is added instead of multiplied, so it is wrong.
• C: This has 2R² and a strange "2r", so it is wrong.
• D: This ignores the other terms, so it is wrong.

Exam tip

When differentiating with respect to one letter in a formula, treat every other letter as an ordinary number. Here, h and r behave just like 5 or 7. Also, keep the constant factor (πh/3) outside and differentiate only the bracket.