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Numerical Derivative Calculator

Finite-difference derivative of a polynomial, sin, or exp. Central, forward, or backward. Not symbolic d/dx. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 9/9 tests · Production surface contract 2/2 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Finite-difference derivative of poly, sin, or exp at x.
Scope
Not symbolic CAS d/dx
Verification
Engine tested · 9/9 tests · Production surface contract 2/2 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
2 golden · 2 boundary · 5 property · Production surface contract 2/2 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Forward(f(x+h) − f(x)) / h
Backward(f(x) − f(x−h)) / h
Central(f(x+h) − f(x−h)) / (2h)
iNot symbolic differentiation. Polynomial coefficients are constant-first. Default h = 10⁻⁶. Degree ≤ 8.

How to use

1

Choose difference and f

Poly needs coeffs. sin and exp are f(x)=sin x and eˣ. Central is usually more accurate.

2

Enter x and optional h

The exact derivative is shown beside the finite-difference approximation.

Example calculations

Common configurations with formula and result.

ϟ

x² at 3

coeffs 0,0,1, central

f′ = 2x
6
ϟ

sin at 0

central

cos 0
1

Numerical Derivative calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
f′(x) is approximated by a forward, backward, or central difference with step h. Polynomial, sin, and exp have an exact derivative used as a check — not a CAS.
What it calculates
Finite-difference derivative of poly, sin, or exp at x.
Inputs
  • method
  • integrand
  • x
  • h?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
Formula
forward / backward / central difference
Assumptions
  • Not symbolic CAS d/dx
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • h > 1 → VALUE_OUT_OF_RANGE
  • unknown method (including autodiff) → INVALID_MODE
Example
central poly 0,0,1 at x=3 → 6
Validation cases

2 published on this page · 9/9 tests · Production surface contract 2/2 · View evidence

  • method=central integrand=poly coeffs=0,0,1 x=3 → exact=6
  • method=autodiff integrand=sin x=0 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Numerical Interpolation Calculator, Numerical Root Calculator, One-Dimensional Optimization Calculator, Simpson Rule Calculator . Explore all Numerical Calculus.

→ Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → Gauss Quadrature Calculator n-point Gauss–Legendre quadrature of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. › → Golden Section Search Calculator Golden-section search for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not Brent, not multivariate. Runs locally. › → Numerical Interpolation Calculator Piecewise-linear, Lagrange, or natural cubic spline of tabulated (x, y). Not regression, not CAS. Runs locally. › → Numerical Root Calculator Find a real root of a polynomial with Newton, bisection, or secant. One capability, three methods. Not the nth-root page. Runs locally. › → One-Dimensional Optimization Calculator Minimize a polynomial, sin, or exp on [a, b] by golden-section search or Brent. Not multivariate, not max, not CAS. Runs locally. › → Simpson Rule Calculator Composite Simpson 1/3 of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Can this differentiate a symbolic expression?

No. Symbolic d/dx is out of scope. This page is a finite difference of poly, sin, or exp, with an exact check.

Where does this run?

Locally in the browser by default. REST and MCP call the same derivative engine.