Concept preview

What you will be able to do

1. 01Carry out a complete six-step procedure for a related-rates problem: draw and label the changing quantities, record the known and unknown rates, find an equation relating the variables, differentiate it with respect to time, substitute the values true at the instant, and solve with correct units.
2. 02Explain why you must differentiate the relation BEFORE substituting the instant's numbers, and identify the chain-rule factor $\dfrac{d}{dt}$ that ties each geometric variable to time.
3. 03Solve the classic related-rates models — the sliding ladder, the expanding sphere, the draining cone, and the streetlight shadow — and interpret the sign of each rate (increasing or decreasing) in context.

Related-rates problems feel hard only because the numbers arrive out of order and the geometry hides in a word problem. A fixed procedure removes the guesswork. Run these six steps every single time, in this order, and the problem solves itself.