---
title: "The Newton-Raphson Method"
description: "Follow the tangent at the current guess down to the x-axis to get the next guess, which is x - f(x)/f'(x). Note when it fails: a nearly flat tangent throws the next guess far away."
canonical: https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd
source: https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd.md
retrieved: 2026-09-12
---

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# The Newton-Raphson Method

Follow the tangent at the current guess down to the x-axis to get the next guess, which is x - f(x)/f'(x). Note when it fails: a nearly flat tangent throws the next guess far away.

Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd

## Ready when they can

- Apply one step of the formula from a given starting value
- Explain the formula as the x-intercept of a tangent
- Describe a starting value for which the method fails and say why

## Lesson: Sliding down the tangent to the root

Start with a rough guess x0. Draw the tangent to the curve at x0 and take where it meets the x-axis as your next guess x1. The tangent at x0 has equation y = f(x0) + f prime(x0)(x - x0), and setting y = 0 gives x1 = x0 - f(x0)/f prime(x0). Repeat from the new point.

**Example.** For f(x) = x squared minus 2 from x0 = 1, f(1) = -1 and the slope f prime(x) = 2x gives 2. The step is 1 - (-1)/2 = 1.5, already much closer to the root near 1.414. Near a simple root with a good start, each step roughly doubles the correct digits.

A flat tangent never meets the axis, and the formula divides by zero, so a zero derivative breaks the method outright. Bad starts can also overshoot far away or cycle between values without settling. A steep slope gives a confident step while a shallow one gives a wild leap.

**Tip.** Pick the start from a sketch near the root you want, and check each new guess stays nearby. If guesses jump far away or repeat old values, stop and restart closer. Speed means nothing without a start the geometry can work with.

**Recap.** Follow each tangent down to the axis for the next guess, and restart closer whenever the steps misbehave.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Locating Roots and Iterative Methods](https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM)
- [Tangents and Normals to a Curve](https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp)

## Opens up

- [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR)