# Rate of change

The derivative with respect to time, or any other input, tells you how fast the quantity changes. Position gives velocity, and velocity gives acceleration.

Maths (university) · Part 03: Differentiation · Card 3 of 10 · Last updated 2026-10-19

## Analogy

Filling a bath: the water level is the quantity, and how fast it rises is the rate.

## Example

**Position s(t) = 5t^2 metres**

Velocity v(t) = s'(t) = 10t.
At t = 3 s: v = 30 m/s.
Acceleration a(t) = v'(t) = 10.
That is 10 m/s^2, constant.

## In practice

Used in physics, economics and biology wherever something grows or shrinks over time.

## Practice

### Flashcards

- **If s(t) = 5t^2, what is the velocity at t = 3?** 30, since v(t) = 10t.

## Questions people ask

### How are position, velocity and acceleration related by derivatives?

Velocity is the derivative of position with respect to time. Acceleration is the derivative of velocity, so it is the second derivative of position. If position is 5t squared metres, velocity is 10t metres per second and acceleration is 10 metres per second squared.

### What do the units of a derivative tell me?

The derivative's unit is the output unit divided by the input unit. If distance is in metres and time in seconds, the derivative is in metres per second. Checking units is a quick way to see whether your derivative makes physical sense.

## Related concepts

- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)
- [Tangent line](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-2)
- [Power rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-4)
- [Product rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-5)

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