Related rates, step by step

A related-rates problem is a shock moving through a geometric relation. Walk the five steps, then drag the ladder until both rates make sense.

  1. 01

    Name the quantities

    Write down every changing length or volume, the rates you are given, and the rate you want. Related rates use Leibniz form: dh/dt, dV/dt.

  2. 02

    Choose the governing relation

    One equation has to connect those quantities: a known formula, an area or volume, or similar triangles. Pick it before you differentiate.

  3. 03

    Differentiate with respect to t

    Differentiate both sides with respect to time, applying the chain rule to every variable. Do not plug in numbers yet.

  4. 04

    Substitute the known values

    Only after differentiating. Substituting first turns a related-rates problem into a wrong static problem.

  5. 05

    Solve and interpret

    Solve for the unknown rate, keep the units, and say what the sign means. Then drag the scene until the same numbers appear on the readout.

x = 6.0y = 8.0010 ft

Drag the amber handle or focus it and use the arrow keys (Shift for larger steps).

x (ft)
6.0
y = sqrt(100 - x^2)
8.000
dy/dt (ft/s)
-1.5000
Relation
x^2 + y^2 = 10^2 = 100
Differentiate the relation with respect to t
2x\,\frac{dx}{dt} + 2y\,\frac{dy}{dt} = 0
Given rate
\frac{dx}{dt} = +2 \text{ ft/s}
Solve for the unknown rate
\frac{dy}{dt} = -\frac{2x}{2y}\,\frac{dx}{dt} = -\frac{x}{y}\,\frac{dx}{dt}
Substitute the current state
\frac{dy}{dt} = -\frac{2(6)(2)}{2(8)} = -1.5
Ladder base x = 6.0 feet, height y = 8.00 feet, dy/dt = -1.500 feet per second.

Differentiate the relation before you substitute. The relation holds for every instant, so its derivative in t links the rates. If you plug the current numbers in first, x becomes a fixed constant, its derivative is zero, and the equation can no longer solve for how fast the top slides down the wall.

The drag stops at x = 9. Since dy/dt = -2x / sqrt(100 - x^2) and the height y = sqrt(100 - x^2) heads to 0 as x approaches 10, the speed |dy/dt| grows without bound near the wall. A ladder problem that asks for the rate at the instant the ladder is flat has no finite answer, an AP favorite.

Setup trainer: related rates setup. Guide: related rates. Night-before list: AB cram sheet.