# higher-order partial derivatives

(60 minutes to learn)

## Summary

When we take partial derivatives of partial derivatives, we get what are known as higher-order partial derivatives. This can describe local properties of a function which aren't captured by the first-order approximation.

## Context

This concept has the prerequisites:

- partial derivatives
- linear approximation (The differentiability condition is necessary for the mixed partials to be symmetric.)

## Core resources (read/watch one of the following)

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 3.6, "Higher-order partial derivatives," pages 120-124

## Supplemental resources (the following are optional, but you may find them useful)

## -Paid-

→ Multivariable Calculus

An introductory multivariable calculus textbook.

Location:
Section 13.4, "Partial derivatives," subsection "Higher-order partial derivatives," pages 874-875

## See also

- The single-dimensional Taylor approximations can be generalized to the multivariate case .