Oxford Calculus: Partial Differentiation Explained with Examples

University of Oxford Mathematician Dr Tom Crawford explains how partial differentiation works and applies it to several examples.

Maple Learn Worksheet: https://learn.maplesoft.com/index.html#/?d=FGCTESFNDKHLKTBUGQDOLOCOKTCSJNMHEKIQJRLRPQDJKIKMJQDKJMBQFPCQKKEKOKJIJHNONRGGAGHOGLKSDPPGFLIULNFMLSGM

Worksheet solutions: https://learn.maplesoft.com/index.html#/?d=BLIJHGFPALINFKHPHRMPLQPPOJBQAHGQLTCRPQCRFTPIPLCGANAHHJEFHGMPLTBOLRIQBKCTEJBJGMGUKKBTGHCQIHHTEMDRHMGQ

We begin by looking at the limit definition of a normal or ‘full’ derivative which we use to motivate the idea behind partial derivatives. Once the partial x and partial y derivatives are defined, we then calculate them for a polynomial function f(x,y) which defines a 3D surface. This is plotted in the Maple Calculator App to help to visualise the changing behaviour of the function in different directions. You can download the app for free from Google Play and the App Store.

Android: https://play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en
Apple: https://apps.apple.com/us/app/maple-companion/id1466659419

We then look at a more complicated example involving exponential and trigonometric functions, which requires the use of the product rule and the chain rule. You can find more questions for you to work through in the Maple Learn worksheet created by Tom (link above).

Other videos in the Oxford Calculus series can be found here: https://www.youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S

Finding critical points for functions of several variables: https://youtu.be/Leomuu82-u8

Classifying critical points using the method of the discriminant: https://www.youtube.com/watch?v=5M_ts8Q2LEM

Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: https://www.youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ

Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford

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