# Tangent line

The straight line that touches a curve at one point and has the same slope there. Its equation is y - f(a) = f'(a)(x - a).

Maths (university) · Part 03: Differentiation · Card 2 of 10 · Last updated 2026-10-19

## Analogy

A ruler laid against a bend in the road, touching it at just one spot.

## Example

**Tangent to y = x^2 at x = 3**

f(3) = 9 and f'(3) = 6.
y - 9 = 6(x - 3).
So y = 6x - 9.
At x = 3.1 the line gives 9.6; true value 9.61.

## In practice

Tangent lines give quick, good estimates of a function near a point.

## Practice

### 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

### How do I find the equation of a tangent line?

Find the point (a, f(a)) and the slope f'(a). Then write y - f(a) = f'(a)(x - a) and tidy up. For y = x squared at x = 3, the point is (3, 9) and the slope is 6, giving y = 6x - 9. Check: x = 3 gives back 9.

### What is a normal line?

The normal line passes through the same point as the tangent but at right angles to it. Its slope is the negative reciprocal of the tangent's slope. If the tangent slope is 6, the normal slope is -1/6. Normal lines matter in physics, for example when finding the direction a surface pushes back on an object resting on it.

## Related concepts

- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)
- [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)

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