About this tool
The Newton's Method Root Finder is a free, in-browser calculator that solves f(x) = 0 numerically using the Newton–Raphson iteration x_{k+1} = x_k − f(x_k)/f′(x_k). Starting from a seed value x₀, each step follows the tangent line at the current point down to where it crosses the x-axis, and that crossing becomes the next estimate. The derivative f′ is evaluated numerically with a symmetric difference, so any expression you type — polynomials, roots, trig, exponentials, logs — works without you supplying the derivative.
Everything runs locally in your browser with a small expression parser, so nothing you enter is uploaded. Type f(x) using x as the variable (for example x^2 - 2, cos(x) - x, or exp(x) - 3*x), set a starting guess, and optionally adjust the tolerance and the maximum number of iterations. The tool iterates until the step size |x_{k+1} − x_k| or the residual |f(x_k)| falls below the tolerance, or until the iteration cap is reached, and it stops safely if the derivative goes flat (near zero) so it never returns a bogus number.
The result panel reports the converged root, the residual f(root) at that root, and how many iterations were needed, alongside a full iteration table listing k, x_k, f(x_k), and the error at each step. Newton's method converges quadratically near a simple root — the number of correct digits roughly doubles each step — which is why x² − 2 from x₀ = 1.5 nails √2 ≈ 1.414214 in just a handful of iterations.