What is numerical differentiation?

Numerical differentiation estimates the derivative of a function at a point using only function evaluations, without finding a symbolic formula. It is useful when the function is complex or only known at discrete data points.

Which method is most accurate?

Central difference is second-order accurate (error ∝ h²), while forward and backward difference are first-order (error ∝ h). For smooth functions, central difference is almost always the best choice.

Why does making h very small sometimes give worse results?

Very small h causes catastrophic cancellation in floating-point arithmetic. The numerator f(x+h) − f(x) becomes the difference of two nearly equal numbers, amplifying round-off errors. A step around 1e-5 to 1e-7 is typically optimal.

What functions can I enter?

You can use standard operators (+, -, *, /, ^), and functions such as sin, cos, tan, asin, acos, atan, log (natural), log10, log2, exp, sqrt, abs, floor, ceil, and constants PI and E.

What are the formulas used?

Forward: [f(x+h) − f(x)] / h. Backward: [f(x) − f(x−h)] / h. Central: [f(x+h) − f(x−h)] / (2h).

Numerical Derivative Calculator

Name: Numerical Derivative Calculator
Availability: InStock
Author: Nham Vu

Compute approximate derivatives at a point using forward, backward, and central finite difference formulas with adjustable step size h.

Function Input

f(x) =

Supports: +, -, *, /, ^, sin, cos, tan, log, exp, sqrt, abs, PI, E

x (point)

Step size h

Quick h presets

Formulas

Fwd: [f(x+h) − f(x)] / h

Bwd: [f(x) − f(x−h)] / h

Ctr: [f(x+h) − f(x−h)] / (2h)

Enter a function and click Calculate

Function Evaluations

f(x−h)

—

f(x)

—

f(x+h)

—

Derivative Approximations at x = —

Forward Difference

[f(x+h) − f(x)] / h

Order: O(h)

—

Backward Difference

[f(x) − f(x−h)] / h

Order: O(h)

—

Central Difference ★

[f(x+h) − f(x−h)] / (2h)

Order: O(h²) — most accurate

—

Agreement Between Methods

|Fwd − Ctr|

—

|Bwd − Ctr|

—

|Fwd − Bwd|

—

Central Difference vs Step Size h

h	Central f′(x)

Summary

Compute approximate derivatives at a point using forward, backward, and central finite difference formulas with adjustable step size h.

How it works

The tool parses the function expression entered by the user into a safe JavaScript evaluator.
It computes f(x), f(x+h), and f(x−h) using the given point and step size.
Forward difference: [f(x+h) − f(x)] / h approximates f'(x) with O(h) error.
Backward difference: [f(x) − f(x−h)] / h also has O(h) error.
Central difference: [f(x+h) − f(x−h)] / (2h) is more accurate with O(h²) error.
Results are displayed with up to 10 significant digits for comparison.

Use cases

Verify analytical derivatives by comparing with numerical estimates.
Approximate derivatives of functions that are difficult to differentiate by hand.
Study the effect of step size on finite difference accuracy.
Learn and teach numerical differentiation methods in calculus courses.
Quick sanity-check during scientific computing or engineering work.
Explore error behavior (truncation vs. round-off) as h varies.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu