Section 13.4

Higher Order Partial Derivatives

Taking derivatives of derivatives. Does order matter?

1

Introduction

Just like tells us about concavity, second partial derivatives tell us about the curvature of a surface.

There are four second-order partials for :

  • : Differentiate wrt , then again.
  • : Differentiate wrt , then .
  • : Differentiate wrt , then .
  • : Differentiate wrt , then .
2

The Derivative Tree

Notice the middle two? and are "Mixed Partials".

3

Clairaut's Theorem

Theorem

If and are continuous, then:

The order of differentiation usually doesn't matter!

4

Worked Examples

Example 1: Basic Computation

Find for .

1. First Partials: , .

2. Second Partials:

Example 2: Verifying Clairaut's Theorem

Let . show .

Path 1: x then y

  • 1.
  • 2. Differentiate wrt y:


Path 2: y then x

  • 1.
  • 2. Differentiate wrt x:


They match perfectly!

Example 3: Third Order Partial

Find for .

1. .

2. (Product Rule).

3. Differentiate wrt x again:



.

Answer: .

5

Practice Quiz

Loading...