Newton’s method
Improve an estimate of a root by following the tangent line to the axis.
Newton’s method is one of 1 numerical methods formula in the calculus section of this library, and it is used at ap · university level.
Why newton’s method works
Replace the curve by its tangent line at the current guess and solve that easy linear equation instead. The tangent crosses the horizontal axis at the current guess minus the function value over the slope, and when the tangent is a good stand-in for the curve, that crossing sits nearer the true root.
What each symbol means
$x_n$ is the current estimate.
Newton’s method: when it holds
$f'(x_n)\ne0$; convergence depends on the starting value and local behavior.
When it stops applying
A flat spot ruins it. If the slope at the current guess is 0 the tangent never meets the axis, and a near-zero slope hurls the next guess into the distance. Starting at 0 for the cubic x³ − 2x + 2 makes the iteration bounce between 0 and 1 forever without ever converging.
Newton’s method: a worked example
For $x^2-2=0$ and $x_0=1.5$, $x_1=1.5-0.25/3\approx1.4167$.
The mistake to avoid
What people do: Flipping the fraction and subtracting the slope divided by the function value.
Why it goes wrong: The correction then blows up precisely when you are closest to the root, because a tiny function value in the denominator throws the next guess far away.
Do this instead: Put the function value on top and the slope underneath, then subtract that step from the current guess.
Newton’s method: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x_n$ is the current estimate.
- Check the conditions before substituting. $f'(x_n)\ne0$; convergence depends on the starting value and local behavior.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Numerical methods
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where newton’s method comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about newton’s method
How many steps does it normally take?
Near a simple root the number of correct digits roughly doubles each pass, so three or four passes usually reach the limit of a calculator display.
How should I pick the starting guess?
Sketch the function or hunt for a sign change, then start near there. Avoid places where the graph flattens out, since those throw the first step off course.
What happens at a repeated root?
Progress slows to a crawl. The slope approaches 0 alongside the function value, so the fast doubling of correct digits is lost.
How good is the first estimate for the square root of 2?
Starting at 1.5 gives about 1.41667, already correct to two decimals, and one more pass gives 1.4142157 against the true 1.4142136.