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 .