# Derivative

The derivative of f at a is the limit of (f(a + h) - f(a)) / h as h tends to 0. It measures the instantaneous rate of change of f at a.

Maths (university) · Part 03: Differentiation · Card 1 of 10 · Last updated 2026-10-19

## Analogy

A speedometer: it shows your speed right now, not your average over the trip.

## Example

**f(x) = x^2 at x = 3**

(f(3+h) - f(3))/h = (9 + 6h + h^2 - 9)/h.
This simplifies to 6 + h.
As h tends to 0, it tends to 6.
So f'(3) = 6.

## In practice

Derivatives tell you how steep a curve is, how fast a quantity grows, and where it peaks.

## Practice

### Flashcards

- **Give the limit definition of f'(a).** The limit as h tends to 0 of (f(a + h) - f(a)) / h.

### Interview questions

**Find the equation of the tangent line to y = x^2 at x = 3, and explain what the derivative represents.**

- f(3) = 9 gives the point (3, 9)
- f'(x) = 2x so the slope is 6
- Tangent line y - 9 = 6(x - 3), so y = 6x - 9
- Derivative is the instantaneous rate of change

## Questions people ask

### What does the derivative actually measure?

It measures how quickly the output changes when the input changes by a tiny amount. Graphically, it is the slope of the tangent line at a point. For a car's position, the derivative is its speed at that exact moment.

### Why is h in the definition of the derivative?

The number h is a small step from a to a + h. The fraction is the slope of a line through two nearby points. As h shrinks to zero, that line turns into the tangent line, and its slope becomes the derivative.

## Related concepts

- [Tangent line](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-2)
- [Rate of change](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-3)
- [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)

Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1
Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-3-concept-1