Newton-Raphson Single Step Calculator

Enter a function f(x), its derivative f'(x), and the present value x_n. The calculator evaluates both expressions at that point, divides the function value by the derivative, and subtracts the resulting correction to produce x_(n+1). It also returns every intermediate number, making the result useful for checking homework, documenting a numerical workflow, or inspecting one iteration before deciding whether to continue.

Enter the function, derivative, and current estimate

Start by writing the function whose root you want to approximate and a separate expression for its derivative. Both expressions use x as the variable. Arithmetic operators, parentheses, powers, constants pi and e, and common functions such as sin, cos, tan, exp, log, sqrt, and abs are supported. Write multiplication explicitly, as in 2*x rather than 2x. For the classic square-root example f(x) = x^2 - 2, enter x^2 - 2 for the function and 2*x for the derivative. Then supply x_current, the estimate at which this one iteration begins. A good initial estimate is normally near the root and lies in a region where the derivative exists and is not zero. The calculator accepts optional prefixes such as f(x) = and f'(x) =, but plain expressions are the clearest choice for scripts and copied calculations. Values are treated as dimensionless numbers, so if the function represents a physical quantity, keep the units consistent throughout the expression and estimate. The parser handles mathematical syntax only and never evaluates program code. Expressions are length-bounded, and every supplied or computed numeric value must be finite. These restrictions make a saved input reproducible and prevent an apparently successful update from hiding an undefined operation.

Follow the single Newton-Raphson update

Newton-Raphson uses the tangent line at the current estimate to predict where the graph crosses the x-axis. The calculator first evaluates f(x_current) and f'(x_current). It then forms the correction f(x_current) / f'(x_current) and applies x_next = x_current - correction. The output exposes the current estimate, function value, derivative value, correction, and next estimate, so each part of the arithmetic can be reviewed independently. For x^2 - 2 at x_current = 1.5, the function value is 0.25 and the derivative value is 3. Dividing gives a correction of about 0.0833333333, which is subtracted to obtain about 1.4166666667. Only one update is performed: the returned x_next is not fed back into the formula automatically. That deliberate boundary makes this capability suitable for lessons, step-by-step worksheets, controlled pipelines, and algorithms in which another system decides when to stop. The precision option changes only how many decimal places appear in the JSON result. Expression evaluation and the subtraction use JavaScript floating-point values before presentation rounding, so choosing fewer displayed places does not cause a sequence of hidden intermediate roundings within this step. If the derivative evaluates to zero, or either expression is undefined at the current point, the operation stops with an input error rather than returning an infinite or misleading estimate.

Interpret the next estimate and decide what follows

The next estimate is a numerical proposal, not proof that a root exists or that convergence will occur. Newton-Raphson often converges rapidly when the starting point is sufficiently close to a simple root and the function is smooth there, but a single step cannot establish those conditions. Inspect the magnitude of the function value and correction. A small correction can suggest that the estimate is stabilizing, yet it may also occur where the derivative is very large, so evaluate f(x_next) in a later step before declaring success. Conversely, a large correction warns that the tangent line crosses the axis far from the current point. Repeated iterations may diverge, oscillate, enter a region where the function is undefined, or converge to a different root than intended. Multiple roots also reduce the method's usual convergence speed. For dependable work, define a stopping rule based on both the correction and residual, impose a maximum iteration count, and compare against a bracketing method when the initial interval is known. This calculator intentionally does not estimate the derivative numerically; the derivative expression is explicit, reviewable, and evaluated at exactly the same x_current as the function. That design is helpful when auditing symbolic differentiation, teaching tangent-based root finding, or integrating one deterministic update into a larger solver that owns convergence checks and iteration limits. The API costs $0.002 per request, while the browser version can run the same pure calculation locally.