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.

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.

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).

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[thevastuniverse]

You are not only a Oxford professor but most importantly, you are a fantastic math teacher! I love your videos. :)

[Moonferret]

This guy is a fantastic teacher. A lot of teachers make things as complicated as possible, possibly to keep an air of superiority over those they are teaching. He keeps it as simple as possible, in order that if people can learn it, they will.

[amiruddinarif]

After watching this, partial differentiation doesn't seems as intimidating as before! Thanks Tom!

[rrodann]

Being an engineering freshman with weak basics of derivatives (online classes lul) I understood everything. Thank you Dr Tom. God bless

[tashdid]

The least math professor looking math professor. Absolutely Brilliant!

[secyll]

In the first 4 minutes you've already taught me more about deriving functions than my math teacher did in a whole year of school. So many people would be interested in computer science and mathematics if they would have had great professors earlier in their lives.

[keithrobinson]

Desperately need more maths teachers like Tom.

[CelloSeanMuller]

This is fantastic! After nearing the end of my semester of calculus 2, I was a bit nervous about multivariable calculus next fall, but this introduction to partial differentiation was very logical and much more approachable than I would have expected! Thank you for the wonderful content Tom!

[davidforeman]

As someone who struggled to get the old Oxford GCE at grade 4 some 50 years ago (ie before you were born), but needs to be able to manipulate Karl Friston's Free Energy Principle, I am utterly grateful for your enthusiasm and clarity.

[andreeacatalina]

I tried for the past 4 hours to understand these, and with this video I understood every in 20 minutes. Now that's a great teacher.

[Jeremymautino]

Tom, I'll tell you that I'm a pre-college student who so far has only seen algebra up to the topic of limits (I'm not in college yet and haven't seen calculus), but I understood this video 100%. you really make it very easy

[CGKittenz]

First time I clearly understood the implications of partials and not just the steps. Thank you!

[mikesbasement]

You just described this topic more clearly than an entire semester at university, and did it in 18 minutes. Damn. Good show man :)

[varshiniganesh]

i'm a freshman doing an engineering degree and i just spent the last ten minutes struggling with other channels telling me about graphs and i was sooo close to giving up before i found this video, so thank you so much, your straight forward teaching method helped me a lot

[johngardner]

I'm a teacher of Mathematics and I can say that that lesson was exceptionally well explained !

[exoticcoder]

Wow This is the clearest explanation I have come across, I understand every word you said, I grabbed the concept within 18 mins and it's one of the most meaningful 18 mins in my life, jusy by listening to how you talk is already very healing for me, not to mention the precise content, Thank you very much !

[shaanveersamra]

This is such a good intro to partial differentiation, your presentation is very easy to follow

[MrRadbadger]

Tom is as skilled in the art of teaching as he is in maths. 10/10👍

[archivist]

I did my Maths A Levels in 1983, but this was a fun way to revise. You remind me of my Pure Maths teacher Mr Hobson (a Cambridge graduate, not Oxford), and wished at the time that everyone could be taught by someone with his enthusiasm for playing with numbers so they could overcome that silly 'maths is hard and boring' attitude that kids hear from other kids and adults too. And now, thanks to the internet, and your YouTube channel in particular, everyone can have that sort of learning experience.

Keep doing what you do.

[botofwar]

never knew that such an app like maple calculator existed on phones.... thank you