AP Calculus AB · Unit 4: Contextual Applications of Differentiation

4.5 Solving Related Rates Problems

The steps

1) Draw and label. 2) Write the equation relating the variables. 3) Differentiate with respect to t. 4) Substitute known values and rates. 5) Solve for the unknown rate.

Order matters

Substitute specific numbers after differentiating — never before. Plugging in a changing quantity too early treats it as a constant (its rate would wrongly vanish). Sliding-ladder problems often use the Pythagorean theorem as the relating equation.

Key terms

Related-rates steps: Draw, relate, differentiate in t, substitute, solve.
Substitute last: Plug in numbers after differentiating, not before.
Why substitute last: Early substitution turns a variable into a constant.
Ladder problems: Use the Pythagorean theorem to relate the sides.
Final step: Solve for the unknown rate.

Try an AP-style question

A 13-ft ladder leans on a wall. Its base slides away at 2 ft/s. How fast is the top sliding down when the base is 5 ft from the wall?

5/6 ft/s
2 ft/s
12/5 ft/s
10/13 ft/s

Show the answer

A. Top height y = 12. 2x x' + 2y y' = 0 → 5(2) + 12y' = 0 → y' = −5/6 (the top moves down).

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