Short on time? This page is the recap. For explanations, the diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- Related rates: quantities linked by an equation change together, so their rates are linked too.
- All quantities depend on the same independent variable, time t. Differentiate everything with respect to t.
- The chain rule is the basis: d/dt of g(x) is g′(x) · dx/dt.
- When two changing quantities are multiplied or divided, you also need the product rule or the quotient rule.
- A decreasing quantity has a negative rate.
Key relationships
| Expression | d/dt |
|---|---|
| xⁿ | n·xⁿ⁻¹ · dx/dt |
| x² + y² | 2x · dx/dt + 2y · dy/dt |
| xy | x · dy/dt + y · dx/dt |
| x/y | (y · dx/dt − x · dy/dt)/y² |
| πr²h (both r and h change) | 2πrh · dr/dt + πr² · dh/dt |
| πr²h (r fixed) | πr² · dh/dt |
Order of work: (1) equation true at all times; (2) differentiate with respect to t; (3) substitute instant values; (4) solve and give units.
Assumptions behind the method
- Every quantity in the equation is a differentiable function of t.
- The equation holds at all times near the instant, not just at that instant. Only then can you differentiate it.
- A quantity is treated as a constant only if it never changes. A value “at the moment when…” is not a constant.
Mistakes to avoid
- Substituting first. It turns variables into constants and their rates into 0.
- Missing rate factors: writing 3s² instead of 3s² · ds/dt.
- “Product rule” as rate × rate: d/dt(xy) ≠ (dx/dt)(dy/dt).
- Wrong sign: “shrinking at 2 cm/s” means a rate of −2 cm/s.
- Quotient rule order: the top is (bottom × rate of top) − (top × rate of bottom).
- Mixed units: convert metres and centimetres before substituting.
Quick self-check
- y = x² + 1, x = 3 and dx/dt = 4. Find dy/dt. (dy/dt = 2x · dx/dt = 24)
- A = πr², r = 5 cm and dr/dt = 2 cm/s. Find dA/dt. (2πr · dr/dt = 20π cm²/s)
- P = xy, x = 2, y = 7, dx/dt = 3 and dy/dt = −1. Find dP/dt. (x · dy/dt + y · dx/dt = −2 + 21 = 19)
Next: practice questions.