How to Solve Related Rates Problems in Calculus
Related rates problems give you the rate of change of one quantity and ask for another. Almost every one follows the same 5 steps.
Quick answer
1. Draw. 2. Identify variables and what’s given vs. wanted. 3. Find an equation that relates them. 4. Differentiate implicitly with respect to time. 5. Plug in values and solve.
Worked example: sliding ladder
A 10 ft ladder leans against a wall. The bottom slides away at 2 ft/sec. How fast is the top sliding down when the bottom is 6 ft from the wall?
Step 1 — Draw
Right triangle: wall (vertical), ground (horizontal), ladder (hypotenuse).
Step 2 — Variables
x = distance of bottom from wall. y = height of top. Given: dx/dt = 2. Find: dy/dt when x = 6.
Step 3 — Relate
Pythagorean: x² + y² = 100.
Step 4 — Differentiate with respect to t
2x(dx/dt) + 2y(dy/dt) = 0.
Step 5 — Plug in
When x = 6, y = √64 = 8. So 24 + 16(dy/dt) = 0 → dy/dt = −1.5 ft/sec.
The negative means the top is moving DOWN. Always interpret the sign.
Common mistakes
- Plugging numbers before differentiating.
- Forgetting the chain rule on each variable.
- Not interpreting the sign.