How to use the Newton's method calculator
Newton's method finds a solution numerically when no closed formula exists. It draws the tangent line at the current point, takes where that line meets the x-axis as the next candidate, and repeats xโโโ = xโ โ f(xโ)/f'(xโ). Enter a polynomial for f(x) and a starting value xโ, and the tool derives f'(x) automatically before running the iteration.
The important part is that the method does not always succeed. A derivative near zero makes the division blow up, so this calculator stops and explains itself when |f'(xโ)| falls below 1e-14. A poor starting value can also oscillate between two points or diverge, so the run is capped at 50 iterations; if |xโโโ โ xโ| < 1e-10 is not met within that limit, the result is reported as not converged.
The output shows f(x) and its derivative in standard form, whether the run converged, the approximate root, f at that root, the iterations used, and both the tolerance and the iteration cap. The table below keeps xโ, f(xโ) and the next candidate xโโโ for every step so you can watch the convergence. Intermediate values are never rounded; full precision is carried through and only the display is cut to 10 significant digits. Polynomials are supported, while trigonometric, logarithmic and exponential functions are not.
Frequently Asked Questions
It draws the tangent line at the current point and takes where that line crosses the x-axis as the next candidate, repeating xโโโ = xโ โ f(xโ)/f'(xโ). Near a root it converges very quickly.
If the derivative is near zero the division blows up, and a poor starting value can oscillate between two points or diverge. This tool stops after 50 iterations and says so in the result.
Polynomials only, because the derivative can then be read straight off the coefficients. Expressions with sin, log, exponentials or parentheses are not supported.