Newton's Method Calculator

Newton's method finds where a function equals zero by repeatedly following tangent lines. From a starting guess, it moves to where the tangent crosses the x-axis, which is usually much closer to the root. This calculator differentiates the function for you, lists every iteration and draws the tangent steps.

Try:

How to use the newton's method calculator

  1. Type the function, such as x^2 - 2.
  2. Enter a starting guess near the root.
  3. Press Calculate to see the iterations and the graph.

Formula

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Worked example: √2 from x² − 2, start 1

  • f(x) =: x^2 - 2
  • Starting guess x₀: 1

◐ Partly checked

x≈1.41421356237x \approx 1.41421356237
Iterations
6
f(x) at the root
−4.44×10−16-4.44 \times 10^{-16}
0.511.52−2−11234x₀x₁x₂x₃rootf(x)
Each tangent line (orange) hits the x-axis at the next, better guess.
Step-by-step working (2 steps)

  1. Newton’s formula Newton's method

    Follow the tangent line at the current guess down to the x-axis.

    xn+1=xn−f(xn)f′(xn),f′(x)=2xx_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)},\quad f'(x) = 2x

  2. Iterate

    Repeat until the guesses stop changing.

    x0=1x1=1.5x2=1.41666666667x3=1.41421568627x4=1.41421356237x5=1.41421356237x6=1.41421356237\begin{gathered}x_{0} = 1\\x_{1} = 1.5\\x_{2} = 1.41666666667\\x_{3} = 1.41421568627\\x_{4} = 1.41421356237\\x_{5} = 1.41421356237\\x_{6} = 1.41421356237\end{gathered}
Formulas used
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
How this was checked
  • ✓ Substituting the root gives f(x) ≈ -4.44e-16, which is 0 to rounding.

Newton’s method finds one root near the starting guess, rounded to 12 significant figures.

Frequently asked questions

How fast does Newton's method converge?

Near a simple root the number of correct digits roughly doubles every step.

Why might it fail?

If the tangent is flat (f′ = 0) or the guess is far from a root, the next guess can jump away. Try a different starting value.

How do I solve cos x = x?

Rewrite it as cos(x) − x = 0 and start from 1. The root is about 0.739085.

Related calculators

Derivative Calculator

Differentiate any function with every rule shown: power, product, quotient and chain rules.

Graphing Calculator

Plot up to three functions with an animated graph.

Cubic Equation Solver

Solve cubic equations: exact roots via the rational root theorem and factor theorem, complex roots, or accurate decimals.

Related guides