---
title: "Derive Newton-Raphson formula"
subject: "Numerical Method"
chapter: 2
key: "semester_4/numerical_method:newton-raphson-derive"
source: https://www.bctengineer.com/q/semester_4/numerical_method/newton-raphson-derive
site: BCT Engineering
---

# Derive Newton-Raphson formula

Numerical Method · Chapter 2: Solutions of Nonlinear Equations

- Asked in: 2072 Asoj, 4 marks; 2071 Magh, 4 marks (asked 2 times)
- Video: [Newton Raphson Method: Derivation](https://www.youtube.com/watch?v=nRbsfnUftT4) — numericalmethodsguy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/newton-raphson-derive
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-2

## Question

Derive the Newton-Raphson iterative formula for solving a non-linear equation (using Taylor series).

## Short answer

Assuming root x0+h, Taylor expansion gives f(x0)+h f'(x0)≈0, so h = -f(x0)/f'(x0), leading to x1 = x0 - f(x0)/f'(x0). Repeated application yields x(n+1) = x(n) - f(x(n))/f'(x(n)).

## Answer

The Newton-Raphson method finds a root of f(x) = 0 by repeatedly replacing the curve by its tangent line.

#### Derivation (Taylor series)
Let *x0* be an approximation to the root and let the exact root be *x0 + h*, where h is a small correction. Then f(x0 + h) = 0. Expand by Taylor's series:

```text
f(x0 + h) = f(x0) + h f'(x0) + (h²/2!) f''(x0) + ... = 0
```

Since h is small, neglect h² and higher powers:

```text
f(x0) + h f'(x0) ≈ 0
h = − f(x0) / f'(x0)
```

The improved approximation is x1 = x0 + h, so

```text
x1 = x0 − f(x0) / f'(x0)
```

Repeating the process gives the general formula:

```text
x(n+1) = x(n) − f(x(n)) / f'(x(n)),     n = 0, 1, 2, ...
```

with f′(x(n)) ≠ 0. Iteration stops when |x(n+1) − x(n)| is less than the required tolerance.

#### Geometrical meaning
The tangent to y = f(x) at the point (x(n), f(x(n))) cuts the x-axis at x(n+1). The slope of the tangent is f′(x(n)) = f(x(n)) / (x(n) − x(n+1)), which gives the same formula.

The method has quadratic convergence: the error in each step is roughly the square of the error in the previous step.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Taylor derivation writes the exact root as:
   A. x0 + h with f(x0+h) = 0
   B. x0 * h
   C. x0 - f(x0)
   D. h/x0
   Answer: A. h is a small correction to the guess. (Key point: Start)

2. To get h, the derivation:
   A. Keeps all terms
   B. Neglects h^2 and higher terms
   C. Neglects f(x0)
   D. Sets f'(x0) = 0
   Answer: B. This gives f(x0) + h f'(x0) = 0. (Key point: Approximation)

3. The resulting iteration is:
   A. x(n+1) = x(n) + f'(x(n))/f(x(n))
   B. x(n+1) = x(n) - f'(x(n))/f(x(n))
   C. x(n+1) = x(n) - f(x(n))/f'(x(n))
   D. x(n+1) = f(x(n))/f'(x(n))
   Answer: C. h = -f/f', so x1 = x0 + h. (Key point: Formula)

4. Geometrically, x(n+1) is:
   A. Where the chord cuts the axis
   B. The midpoint
   C. The curve's maximum
   D. Where the tangent at x(n) cuts the x-axis
   Answer: D. Tangent slope f' leads to the same formula. (Key point: Geometry)

## Related in Numerical Method

- [Limitations of Newton-Raphson](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alimitations-newton-raphson) — Numerical Method, chapter 2
- [Newton-Raphson algorithm](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-raphson-algorithm) — Numerical Method, chapter 2
- [Derive derivatives from Newton forward](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aderive-derivatives-newton-forward) — Numerical Method, chapter 5
- [Derive linear least squares](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aleast-squares-linear-derive) — Numerical Method, chapter 4
- [Iterative method: 4 equations](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aiterative-method-four-equations) — Numerical Method, chapter 3, asked 2×
- [Secant method: explain and derive](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Asecant-method-explain-derive) — Numerical Method, chapter 2, asked 3×

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