Numerical Derivative Calculator

What does this numerical derivative calculator compute?

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.

The published formula or method is forward / backward / central difference, or two-sided [f(x+h)+f(x−h)]/2. The following references support the stated mathematics or method:

• NIST DLMF §3.4 — Differentiation: Numerical differentiation using differences, including truncation and rounding considerations.
• Gilbert Strang, Calculus — Chapter 2: Derivatives (MIT OpenCourseWare): Two-sided and one-sided limits, continuity, and functions whose limits do not exist. Finite numerical samples alone are not proof of a mathematical limit; CalculatorX’s sampling limitations are stated separately. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

A finite step gives a numerical estimate. Very small steps can amplify roundoff, while large steps introduce truncation error; finite samples alone do not prove an arbitrary limit.

The worked example uses coeffs 0,0,1, central with f′ = 2x. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.derivative v1.1.4 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Input precision, finite-step meaning and privacy

Use finite decimal numbers or decimal text for x and h. Nonzero decimal underflow is rejected, never changed to zero. Missing h uses 0.000001; an explicitly empty h remains an error. h must satisfy 0<h≤1. Polynomial input is 2–9 constant-first coefficients, degree at most8; constants use c0,0. Method/function/coefficient aliases must agree. Two-Sided Limit locks method=limit.

The primary value is the requested finite-step approximation, not the analytic derivative. Polynomial x±h uses mathematical sums of the supplied binary64 values with exact dyadic intermediates; stable sin/exp identities also retain h when a binary64 node would round to x. A finite sample does not prove an arbitrary limit. Nonzero primary value/baseline underflow and nonfinite probes/results are rejected. Probe, gap and signed error diagnostics are rounded binary64 and may quantize to zero; subnormal outputs have coarse relative precision.

The historical exact field is a binary64 analytic baseline. abs_err is the signed difference of the published approximation and baseline, not an absolute error or certified bound. Values publish12 significant digits; this does not guarantee12 digits of accuracy for arbitrary h. The complete result includes method/function, x/h, polynomial coefficients, value/baseline/error/formula, and left/right/gap for limit. At unsupported graphic scales a figure may be omitted while the numeric result is retained.

Ordinary form calculations run locally without an account. JSON and CSV retain full inputs, results and calculation version. A shared query URL sends its parameters to the server when opened and may appear in server or third-party logs. Workspace fragments are recomputed from inputs; stored result claims are ignored.