# Related rates

When two or more quantities change with time and are linked by an equation, differentiate the equation with respect to time to link their rates.

Maths (university) · Part 04: Applications of derivatives · Card 6 of 9 · Last updated 2026-10-19

## Analogy

A balloon being inflated: radius and volume grow together, and one rate fixes the other.

## Example

**10 m ladder, base 6 m from wall**

Height y = 8 m, since 6^2 + 8^2 = 10^2.
x^2 + y^2 = 100, so 2x x' + 2y y' = 0.
Base slides out at x' = 2 m/s.
y' = -(6)(2)/8 = -1.5 m/s.

## In practice

Used for sliding ladders, filling tanks and shadows that change length.

## Practice

### Flashcards

- **Why differentiate before substituting in related rates?** Changing quantities must keep their rates until the equation is differentiated.

## Questions people ask

### How do I start a related rates problem?

Draw a diagram and name every changing quantity. Write an equation that holds at every moment, such as Pythagoras for a ladder. Differentiate with respect to time using the chain rule. Only after that, substitute the known numbers for the moment in question.

### Why not substitute the numbers before differentiating?

Quantities like the ladder's height change with time, so substituting a single value turns them into constants and removes their rates. Differentiate the general equation first, then put in the values at the instant you want. Only fixed lengths may be substituted early.

## Related concepts

- [Critical point](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-1)
- [First derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-2)
- [Second derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-3)
- [Mean value theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-5)

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