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Numerical Derivative Calculator

Finite-difference derivative of a polynomial, sin, or exp, plus a numeric two-sided limit. Central, forward, backward, or limit. Discovery /two-sided-limit. Not symbolic d/dx. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published methods
Assurance
Core
Declared partition coverage
PASS · 7/7 declared partitions (tiny-step, subnormal-exp, central, forward, backward, limit, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Finite-difference derivative, or numeric two-sided limit, of poly, sin, or exp at x, with an engine-linked tangent (or limit) figure.
Scope
The Result Card includes an engine-linked figure of f around x (tangent slope is the published finite difference; two-sided limit draws a horizontal at the probe). Same fnValue as REST/MCP. Not symbolic CAS d/dx.
Verification
Engine tested · Source checked · v1.1.2 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.derivative-elementary-stability
Verification revision
2026-10-07.derivative-elementary-stability · 1/1 property · digest 27f3f8a25a41
Legacy regression
14/14 tests · Production surface contract 4/4
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 3/3 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
28/28 golden · 2/2 CVP boundary · 8/8 invalid · 1/1 property · 11/11 cross-interface · 1/1 cross-calculator · 4/4 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — forward, backward, and central differences
Sources
Evidence
4 legacy golden · 3 legacy boundary · legacy regression suite · 28/28 oracle-backed golden · 8/8 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.2 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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)
Two-sided limit[f(x+h) + f(x−h)] / 2
iNot symbolic differentiation. Not one-sided. Not L’Hôpital. Polynomial coefficients are constant-first. Default h = 10⁻⁶. Degree ≤ 8. Limit discovery /two-sided-limit.

How to use

1

Choose difference or limit, and f

Poly needs coeffs. sin and exp are f(x)=sin x and eˣ. Central is usually more accurate for f′. Two-sided limit is a sibling method.

2

Enter x and optional h

The exact derivative or f(x) (for limit) is shown beside the numeric approximation.

Example calculations

Common configurations with formula and result.

ϟ

x² at 3

coeffs 0,0,1, central

f′ = 2x
6
ϟ

lim x² at 3

method=limit

lim = 9
9

Numerical Derivative calculator specification

Version 1.1.2 · Engine tested

Calculation status

Review policy · Evidence

Definition
f′(x) is approximated by a forward, backward, or central difference with step h. method=limit estimates the two-sided lim_{t→x} f(t). Polynomial, sin, and exp have an exact check — not a CAS.
What it calculates
Finite-difference derivative, or numeric two-sided limit, of poly, sin, or exp at x, with an engine-linked tangent (or limit) figure.
Inputs
  • method
  • integrand
  • x
  • h?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
  • left?
  • right?
  • gap?
Formula
forward / backward / central difference, or two-sided [f(x+h)+f(x−h)]/2
Assumptions
  • The Result Card includes an engine-linked figure of f around x (tangent slope is the published finite difference; two-sided limit draws a horizontal at the probe). Same fnValue as REST/MCP. Not symbolic CAS d/dx.
  • Not one-sided limits
  • Not L’Hôpital
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • exp(x) underflow to zero → NUMERICAL_RANGE_ERROR
  • Non-finite function probes, derivative or error → NUMERICAL_RANGE_ERROR
  • h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • h > 1 → VALUE_OUT_OF_RANGE
  • unknown method (including autodiff, left, lhopital, cas) → INVALID_MODE
Example
central poly 0,0,1 at x=3 → 6
Validation cases

2 published on this page · 14/14 tests · Production surface contract 4/4 · View evidence

  • method=central integrand=poly coeffs=0,0,1 x=3 → exact=6
  • method=autodiff integrand=sin x=0 → INVALID_MODE
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Derivative f′(x) and two-sided limit of an elementary f
  • CRC Standard Mathematical Tables — forward, backward, and central differences
    Supports: Forward (f(x+h)−f(x))/h; backward; central (f(x+h)−f(x−h))/(2h); two-sided probe of f
Calculation version
1.1.2

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Interpolation Calculator, Numerical Root 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. › → Euler Method Calculator Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. 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. › → Nonlinear Equation Calculator Find one approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Uses the Numerical Root engine. 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 an approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Enter a starting point or bracket and inspect the stopping criterion. Runs locally. ›

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 or two-sided numeric limit of poly, sin, or exp, with an exact check.

Is the two-sided limit a second engine?

No. method=limit is on the same math.numerical.derivative engine. Discovery URL /calc/math/two-sided-limit. One-sided and L’Hôpital are INVALID_MODE.

Where does this run?

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