# Newton's method

A way to find roots by repeating x_new = x - f(x)/f'(x). Each step follows the tangent line down to the x-axis.

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

## Analogy

Skiing down a slope: you slide along the tangent until you hit the ground, then repeat from there.

## Example

**Solve x^2 - 2 = 0, start x = 1**

x1 = 1 - (1 - 2)/2 = 1.5.
x2 = 1.5 - (2.25 - 2)/3 = 1.41667.
True sqrt(2) = 1.41421.
Two steps already agree to 2 places.

## In practice

Computers use it to solve equations with no neat formula, converging in a few steps.

## Practice

### Flashcards

- **What is the Newton's method update?** x_new = x - f(x)/f'(x).

## Questions people ask

### Why does Newton's method sometimes fail?

It can fail if f' is zero at a guess, since you divide by zero, or if the starting point is far from the root and the tangent sends you somewhere strange. It can also cycle between values. A good starting guess near the root helps.

### How fast does Newton's method converge?

Near a simple root it roughly doubles the number of correct digits each step. In the square root of 2 example, 1.5 is off by about 0.09, and 1.41667 is off by about 0.002. That speed is why calculators use it.

## 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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