Newton's Method Calculator

This calculator finds a root of a function, a value of x where the function equals zero, using Newton's method, one of the fastest and most widely used numerical techniques in mathematics. Many equations cannot be solved with a neat formula, so we turn to iterative methods that start from a guess and improve it step by step until they home in on the answer. Newton's method, also called Newton-Raphson, is the classic example: from a starting point it follows the tangent line of the curve down to where it crosses the x-axis, uses that as the next guess, and repeats. When it works, it converges remarkably quickly, often doubling the number of correct digits with each step. It powers root-finding inside calculators, computer algebra systems, engineering software and machine learning. This tool lets you apply it to your own function. You enter the function f(x) using ordinary mathematical syntax, a starting guess, and the number of iterations, and the calculator runs Newton's method, returning the root it converges on, the value of the function there, which should be very close to zero, and the number of iterations actually used. It estimates the derivative numerically, so you do not need to provide it. The results update as you type. Use it to solve equations that have no algebraic solution, to check a root, or to learn how iterative methods work. A couple of practical notes: the method needs a reasonable starting guess and can fail or jump to a different root if the guess is poor or the derivative is near zero, so if the result looks wrong, try a different starting point. Functions can use operations like x squared, square roots and trigonometric functions.

Newton's method: next x = x - f(x)/f'(x), using a numerical derivative. Needs a good starting guess; try another if it does not converge. Use JS syntax (x*x, sqrt(x), sin(x)).