Related rates problems feel hard because they're really two problems stacked together: translating a word problem into math, and then differentiating it correctly. Most students can do the differentiation once the equation is set up — the setup is where things fall apart. Here's a method that separates those two steps cleanly.
The 5-step method
- Draw a diagram. Label every quantity that changes with a variable, and every quantity given a number.
- Write down what's given and what's asked as rates (things like dx/dt), not just values.
- Find an equation relating the variables — usually geometry (Pythagorean theorem, area, volume formulas) or a relationship stated in the problem.
- Differentiate both sides with respect to time, using the chain rule on every variable that depends on t.
- Plug in the known values (only after differentiating, never before) and solve for the unknown rate.
Worked example
A 10-foot ladder leans against a wall. The bottom slides away from the wall at 2 ft/s. How fast is the top sliding down when the bottom is 6 feet from the wall?
Let x = distance from wall to base of ladder, y = height of top of ladder. Since the ladder length is fixed: x² + y² = 100.
Differentiate both sides with respect to time: 2x(dx/dt) + 2y(dy/dt) = 0
When x = 6, y = 8 (from the Pythagorean theorem). We're told dx/dt = 2.
2(6)(2) + 2(8)(dy/dt) = 0 → 24 + 16(dy/dt) = 0 → dy/dt = −1.5 ft/s
The negative sign makes sense: the top of the ladder is sliding down, so its height is decreasing.
Common mistakes
Plugging in numbers before differentiating. If you substitute x = 6 into the equation before taking the derivative, you lose the variable you needed to differentiate. Keep everything as variables until after step 4.
Forgetting the chain rule on each variable. Since x and y both depend on t, differentiating x² gives 2x(dx/dt), not just 2x — it's easy to drop the rate term.
Stuck on a calculus problem right now?
Tell us what you're working on and get matched with a tutor who explains the reasoning, not just the answer — free quote by email or WhatsApp.
Get a free quote