Finite Difference Numerical Derivative (Forward O(h) vs. Central O(h²) & Second Derivative) Calculator

Compare O(h) forward difference vs. O(h²) central difference truncation errors.

Central Difference First Derivative f′(x)
24.003 df/dx
Exact Analytical First Derivative 24 df/dx
Forward Difference O(h) Estimate 24.453 df/dx
Central 2nd Derivative f″(x) 18 d²f/dx² (exact 18)
Inputs
⚡ Scenarios:
x
Point x at which to differentiate f(x) = x³ + a·x².
a
Coefficient a in f(x) = x³ + a·x².
h
Finite difference step size h.
Results & Graph
Central Difference First Derivative f′(x)
24.003 df/dx
Exact Analytical First Derivative
24 df/dx
Forward Difference O(h) Estimate
24.453 df/dx
Central 2nd Derivative f″(x)
18 d²f/dx² (exact 18)

Central Difference First Derivative f′(x)

Graph 1 • Finite Difference Derivative • Domain Architecture & Live Telemetry Maps Evaluation Point X, Quadratic Coeff A, Step Size H through Finite Difference Derivative Formula.
Central Difference First Deriv: 24.003 df/dx Finite Difference Derivative • Invariants & Metrics Central Difference First Der 24.003 df/dx Exact Analytical First Deriv 24 df/dx Forward Difference O(h) Esti 24.453 df/dx Central 2nd Derivative f″(x) 18 d²f/dx² (exact 18)
Graph 2 • Central Difference First Derivative f′(x) Across Evaluation Point X Sweep (40% → 160%) Evaluates f′_central(x) = [f(x+h) − f(x−h)] / (2h); f″_central(x) = [f(x+h) − 2f across 7 operating points (current input highlighted).
0 df/dx 27.46 df/dx 54.91 df/dx 6.72 0.800x 11.52 1.20x 17.28 1.60x 24.00 2x 31.68 2.40x 40.32 2.80x 49.92 3.20x
Graph 3 • Joint Sensitivity: Evaluation Point X × Quadratic Coeff A Compares Central Difference First Derivative f′(x) when Quadratic Coeff A is shifted +25% (solid) vs. −25% (dashed).
0 df/dx 21.88 df/dx 43.76 df/dx --- Quadratic Coeff A @ 75% 16.38 -30% 21.42 -15% 27.00 +0% 33.12 +15% 39.78 +30%
Summary: At x=2 (h=0.05), Central Difference O(h²) is 24.0025 (error 0.0025) vs. Forward Difference O(h) 24.4525 (error 0.4525).
Summary: At x=2 (h=0.05), Central Difference O(h²) is 24.0025 (error 0.0025) vs. Forward Difference O(h) 24.4525 (error 0.4525).
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Formula & Mathematical Methodology: Finite Difference Derivative Formula

f′central(x) = [f(x+h) − f(x−h)] / (2h)
f″central(x) = [f(x+h) − 2f(x) + f(x−h)] / h²
f′_central(x) = [f(x+h) − f(x−h)] / (2h); f″_central(x) = [f(x+h) − 2f(x) + f(x−h)] / h²
  • evaluationPointX: Evaluation Point X (x)
  • quadraticCoeffA: Quadratic Coeff A (a)
  • stepSizeH: Step Size H (h)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Finite Difference Derivative

Compare O(h) forward difference vs. O(h²) central difference truncation errors.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: f′_central(x) = [f(x+h) − f(x−h)] / (2h); f″_central(x) = [f(x+h) − 2f(x) + f(x−h)] / h².
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Finite Difference Derivative via f′_central(x) = [f(x+h) − f(x−h)] / (2h); f″_central(x) = [f(x+h) − 2f(x) + f(x−h)] / h².

Frequently Asked Questions

How does the Finite Difference Numerical Derivative (Forward O(h) vs. Central O(h²) & Second Derivative) Calculator compute the solution?

It evaluates f′_central(x) = [f(x+h) − f(x−h)] / (2h); f″_central(x) = [f(x+h) − 2f(x) + f(x−h)] / h² directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

Finite Difference Numerical Derivative (Forward O(h) vs. Central O(h²) & Second Derivative) Calculator: inputs, method, and how to read the result

Compare O(h) forward difference vs. O(h²) central difference truncation errors.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Evaluation Point X: Point x at which to differentiate f(x) = x³ + a·x².
  • Quadratic Coeff A: Coefficient a in f(x) = x³ + a·x².
  • Step Size H: Finite difference step size h.

What the calculation includes

With the default values shown, Central Difference First Derivative f′(x) is 24.003 df/dx. The calculation follows Finite Difference Derivative Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Evaluation Point X from 2 x to 3 x changes Central Difference First Derivative f′(x) from 24.003 df/dx to 45.003 df/dx. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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