Newton's Method Calculator (f(x) only, numerical derivative)
Solve f(x) = 0 with Newton's method. Just enter the equation, initial guess, tolerance, and max iterations - the derivative is approximated automatically by numerical differentiation. Shows the approximate root, f(root), iteration count, error, plus a step table and a graph.
Input
Allowed: + - * / ^ (power), (), x, pi/e, plus functions like sin cos tan exp log ln sqrt abs. Examples: x^2 - 2 / cos(x) - x / exp(x) - 3
Result
Approximate root x ≈
1.4142135624
Converged within the tolerance.
f(root)
4.5102e-12
Iterations
4
Final error |Δx|
2.1239e-6
Iteration steps
| n | xₙ | f(xₙ) | f'(xₙ) | |Δx| |
|---|---|---|---|---|
| 0 | 1 | -1 | 2 | — |
| 1 | 1.5 | 0.25 | 2 | 0.5 |
| 2 | 1.41666667 | 0.006944 | 3 | 0.083333 |
| 3 | 1.41421569 | 6.0073e-6 | 2.833333 | 0.002451 |
| 4 | 1.41421356 | 4.5102e-12 | 2.828431 | 2.1239e-6 |
How it works
- Finds a root of f(x)=0 by iteration. Enter an expression (e.g. x^2 - 2, cos(x) - x, exp(x) - 3), an initial guess x₀, a tolerance, and the maximum number of iterations to get the approximate root.
- You do not need to supply the derivative f'(x). A numerical derivative that estimates the slope from nearby values is used, so only the expression is required.
- Supports the four arithmetic operations, powers (^), parentheses, the variable x, constants (pi, e), and functions such as sin/cos/tan, exp, log (natural log), ln, sqrt, and abs.
- The result depends on your choice of initial guess. Setting x₀ near the root you want gives fast convergence; a distant guess may converge to a different root or diverge.
- The iteration table shows xₙ, f(xₙ), the slope, and the change per step, while the graph lets you see the curve and the location of the approximate root at a glance.
- A smaller tolerance increases precision, but due to rounding error values below a certain point may not be meaningful.
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