Related rates

Math

Definition

A calculus technique for finding the rate of change of one quantity by relating it to other quantities whose rates of change are known. You set up an equation connecting the variables, then differentiate both sides with respect to time.

How It Works

  1. Identify all quantities that change with time and assign variables.
  2. Write an equation relating those variables.
  3. Differentiate both sides with respect to time using implicit differentiation and the chain rule.
  4. Substitute known values and rates into the differentiated equation.
  5. Solve for the unknown rate.

Examples

  • A ladder sliding down a wall — find how fast the base moves outward given how fast the top slides down
  • A balloon being inflated — find how fast the radius grows given the rate air is pumped in
  • Water filling a conical tank — find how fast the water level rises

Related Terms

Implicit differentiation Chain rule Optimization

Study This Concept

Practice related rates with free review games in these units:

AP Calculus AB Unit 4: Contextual Applications of Differentiation