Unit 4: Contextual Applications of Differentiation
Related Rates
Solving related rates problems involving multiple changing quantities.
Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed
When an Equation Ties Two Changing Quantities Together
Definition — Related Rates
Differentiating both sides of an equation with respect to time t — treating every changing quantity as an implicit function of t — relates their rates of change. Substitute known numerical values only after differentiating.
Worked Example — Two Cars Moving Apart
Two cars leave an intersection, one heading east at 40 mph, the other north at 30 mph. How fast is the distance between them increasing after 1 hour? Let x = east distance, y = north distance, z = distance between them: x² + y² = z². Differentiate: x(dx/dt) + y(dy/dt) = z(dz/dt). After 1 hour: x = 40, y = 30, z = √(40² + 30²) = 50 (a 30-40-50 right triangle). With dx/dt = 40, dy/dt = 30: 40(40) + 30(30) = 50(dz/dt), so 1600 + 900 = 2500 = 50(dz/dt), giving dz/dt = 50 mph.
Worked Example — A Conical Sand Pile
Sand pours onto a conical pile at 6 ft³/min; the pile's height always equals its base radius (h = r). Find dh/dt when h = 3 ft. V = (1/3)πr²h = (1/3)πh³ (since r = h). Differentiate: dV/dt = πh²(dh/dt). With dV/dt = 6, h = 3: 6 = π(9)(dh/dt) = 9π(dh/dt), so dh/dt = 6/(9π) = 2/(3π) ≈ 0.212 ft/min.
Tip
Substitute specific numerical values only after differentiating — substituting first treats a still-changing quantity as constant and loses its rate entirely.
Common Mistakes
Plugging in known values before differentiating the constraint equation.
Differentiate the general relationship first; only then substitute the specific values at the instant being analyzed.
Key Takeaways
- A related rates problem differentiates a constraint equation with respect to time to connect two rates of change.
- Every quantity changing with time picks up its own rate-of-change factor via the chain rule.
- Numerical substitution happens only after differentiating.
Summary
The next lesson uses the derivative two more ways — to approximate function values locally, and to resolve limits that resist ordinary algebra.
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