LaTeX
f(x+h)\approx f(x)+\nabla f(x)^Th+\frac12 h^T H_f(x)hVariables
- h: displacement vector
- Hf: Hessian matrix
How to use this formula
Approximates a smooth multivariable function near a reference point.
Important notes
- Higher-order terms are omitted in the quadratic approximation.
Quick example
Used in optimization to derive Newton-type methods.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Higher-order terms are omitted in the quadratic approximation.
- For the Multivariable Taylor Expansion, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Check the domain and the convergence, continuity, differentiability, or measurability conditions used by the result.
Worked example
Used in optimization to derive Newton-type methods.
Common mistakes
- When copying Multivariable Taylor Expansion, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Multivariable Taylor Expansion with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Multivariable Taylor Expansion 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+h)\approx f(x)+\nabla f(x)^Th+\frac12 h^T H_f(x)h. - 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 Multivariable Taylor Expansion used for?
Approximates a smooth multivariable function near a reference point.
Can I copy this formula as LaTeX?
Yes. Copy f(x+h)\approx f(x)+\nabla f(x)^Th+\frac12 h^T H_f(x)h 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.