Calculus 12 · Mathematics · Lesson 4.1
Related rates
- Level
- Grade 12
- Reading time
- 14 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Connected rates of change.
What are related rates in calculus?
Short answer
Related rates problems find how fast one quantity changes by relating it to another whose rate is known. You write an equation linking the variables, differentiate both sides with respect to time using the chain rule, and solve for the unknown rate. A classic example is finding how fast a ladder’s top slides down a wall as its base is pulled away.
Concept preview
What you will be able to do
- 01Interpret a related-rates problem as two or more quantities changing in time and linked by an equation.
- 02Apply implicit differentiation with respect to time to relate the rates of change of linked quantities.
- 03Solve a related-rates problem with a systematic procedure: diagram, equation, differentiate, then substitute.
- 04Calculate an unknown rate from given rates and a geometric or algebraic relationship.
- 05Diagnose a flawed related-rates setup, especially substituting numerical values before differentiating.
Many real situations involve two or more quantities that are linked, so that when one changes the others must change too. As a puddle spreads, its radius and its area grow together; as water pours into a tank, the volume and the depth rise together; as a ladder slips, the distance of its foot from the wall and the height of its top change together. A related-rates problem asks how the rate of change of one such quantity is connected to the rate of change of another. The key realisation is that if the quantities are linked by an equation, then their rates of change are linked by the derivative of that equation, and the tool that produces the link is the chain rule applied with respect to time.
School Plan
Related rates
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Related topics
Frequently asked questions
- What is the key step in a related rates problem?
- Differentiating an equation that relates the quantities with respect to time, so each variable’s rate of change appears.
- Why is the chain rule needed?
- Because each variable depends on time, differentiating with respect to time introduces its rate through the chain rule.