LaTeX
f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}Variables
- h: grid spacing
- f: sampled function
- x: center point
How to use this formula
Approximates a second derivative using three equally spaced samples.
Important notes
- The formula is second-order accurate for smooth functions.
- It is a standard building block for finite-difference PDE solvers.
Quick example
The stencil coefficients are 1, −2, 1 divided by h².
Applicability, worked calculation, and verification
Assumptions and domain checks
- It is a standard building block for finite-difference PDE solvers.
- For the Central Difference Second Derivative, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region.
Worked example
The stencil coefficients are 1, −2, 1 divided by h².
Common mistakes
- When copying Central Difference Second Derivative, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error.
Continue the workflow
Use Central Difference Second Derivative in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Central Difference Second Derivative used for?
Approximates a second derivative using three equally spaced samples.
Can I copy this formula as LaTeX?
Yes. Copy f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.