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: learn.maplesoft.com/doc/g1siy...
Worksheet solutions: learn.maplesoft.com/index.htm...
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: play.google.com/store/apps/de...
Apple: apps.apple.com/us/app/maple-c...
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: • Oxford Calculus
Finding critical points for functions of several variables: • Oxford Calculus: Findi...
Classifying critical points using the method of the discriminant: • Oxford Calculus: Class...
Find out more about the Maple Calculator App and Maple Learn on the Maplesoft KZbin channel: / @maplesoft
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collec...

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

[@TomRocksMaths]

Thank you

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Doctor, not professor. :)

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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.

[@TomRocksMaths]

[@rrodann]

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

[@amiruddinarif3194]

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

[@TomRocksMaths]

You're very welcome :)

[@keithrobinson2016]

Desperately need more maths teachers like Tom.

[@CGKittenz]

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

[@TomRocksMaths]

Glad it was helpful!

[@mikesbasement6954]

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

[@secyll5565]

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.

[@TomRocksMaths]

Happy to help :)

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Yes I think the presentation of mathematics is all wrong in schools. I understand there has to be basic levels to get most people up to a functional level, but I really think the creativity and artistry of mathematics is criminally underplayed. Instead of attracting people who are good at and attracted to rigid structure, you could also be bringing in the artists and creatively minded people. As is I think most of these people get completely turned off by the 'follow these steps over and over' approach. More theory / proof / fundamental topics early on would also hook the philosophers, english and even lawyerly types.

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This totally. Teaching is only half knowing the content. The other half is being able to actually teach what you know and that requires, an engaging charismatic teacher. There is a reason someone like Brian Cox was successful on BBC

[@seanmuller8774]

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!

[@TomRocksMaths]

Awesome - thanks Sean :)

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[@davidforeman1]

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.

[@TomRocksMaths]

Thanks David!

[@jacqueslapidieux3182]

Superb! Hadn't thought about partial differentials in almost 30 years (learnt about them in 1st year Econometrics), but your crystal clear explanations and thorough worked examples made everything come flooding back to me! Had to grab a pencil and paper half way through to work out the answers myself. Inspiring!

[@botofwar6567]

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

[@robinhill259]

Clear and concise, delivered at a pace which allows the viewer to grasp the concepts and pause the video if necessary. Well done! 👍

[@clrens8127]

I just love to see people who are passionate about what they teach, especially you Dr. Crawford. The way you teach will definitely incorporate the students'/viewers' curiosity, like myself, which changed my view on calculus from being hard to somewhat fun. Thank you!

[@TomRocksMaths]

Thanks Claurence!

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What a timely upload, this is exactly what i needed right now! Another great video as always (btw I love the shirt)

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[@shaanveersamra954]

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

[@TomRocksMaths]

Awesome - thanks :)

[@leonidasloquendero]

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

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Exactlyyy sameee

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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 !

[@johngardner2888]

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

[@TomRocksMaths]

Thanks John!

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Youre such a good teacher, i am only mid way through a further maths a-level and i completely understand it

[@TomRocksMaths]

Amazing - thanks :)

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uh i have a question ben, ur taking further so its safe to assume u finished pure, how difficult is further pure in comparison to pure?

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@@ahmedgadeIhaq currently taking fp1 and fp2, i'd say it's worth it if you really love pure over applied and it's not as impossible as some people say but there are definitely tricky parts. Gets easier with practice though

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Please keep making videos like these, you’re so good at explaining complicated ideas!

[@TomRocksMaths]

will do!

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It is always important to explain these things from first principles. You did a wonderful job at that. Thanks.

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

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You have an extraordinary gift for making the complex seem simple.

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Wonderful, nay, excellent maths teacher. He imbues clarity on a topic that befuddles many a student!

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He has the brain, the skills and the material for a strict online course. This man can math to the extreme details like the work of Joseph Edwards, Treatise on Integral Calculus. It's time to get dirty with the theorems, proofs, the details on each of these courses. Anything this man explains, becomes transparent. I would like to see math courses where the professor develops the whole theory in every painstaking detail without omitting the difficulties of the theories. Yes math is fun and of course he is an outstanding profession but I think it's time people understand how complex it is throughout the details. Tensor Calculus, Differential Geometry, Measure Theory, Statistics deriving all distributions and proving all the theorems, Relativity Theory, Mechanics and so on. People are hungry for knowledge. These things are not magical or mysterious. They require a ton of work, hours with pencil and paper in hand and writing the proofs of those theorems. Books are available. It's inspiring finding people like him in the net.

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[@TomRocksMaths]

glad I could help :)

[@glaucusatlanticus4481]

Thank you. This didn’t just explain partial differentiations, but also clarified things regarding clarification in general.

[@TomRocksMaths]

Glad it was helpful!

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I realise that you are born to be a teacher. I wish and appreciate if you lecture on more topics of higher mathematics . Thank you and god bless you.

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I kind of regret not discovering this channel a long time ago. Just by watching this specific short vid I already learned a lot. The way the lesson was presented and explained from the smallest and most crucial details was multiple times better than the way our professer teach in our university. Super informative. I mean it. Thank you ^^

[@TomRocksMaths]

[@user-kx9by7jm8u]

The Core objective of Teaching is making people understand = Hence Proved Thank you Dr.Tom

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Simple and straight to the point. Thanks a million, prof !

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You explain this so clearly and with wonderful pace! I wish you were teaching when I was in school 40 years ago. Thanks for the videos.

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I feel the same. Over the years I've gotten better at Maths through self-study. Teachers like Tom would have helped greatly.

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Bro saved me the trouble of goin thru old notes. Much thanks.

[@larianton1008]

Best explanation thus far! Thank you Tom!

[@gandalfthegrey2777]

awesome explanation, just came back from uni and wanted to understand this chapter

[@ottosparky]

Very well explained. The concepts are lucid and easily understood. Best video on explaining the concept of limit and partial differentiation. Thank you!!!

[@yolanankaine6063]

I love your approach to teaching. Very clear and detailed explainations, taking us step by step. Excellent.

[@TomRocksMaths]

Glad it was helpful :)

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Though i knew most of it already, its just amazing to see the enthusiasm !!

[@TomRocksMaths]

Thanks Prasoon - glad you enjoyed it!

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Coolest math teacher ever

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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 KZbin channel in particular, everyone can have that sort of learning experience. Keep doing what you do.

[@TomRocksMaths]

This has made my day

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This is so true I’m in 12th grade (United states so we don’t get exposure to most of these while we’re young) I hated math for the longest but After having an IB geometry teacher it’s easily my favorite subject and I enjoy learning it. Hopefully I can head off to college and study EE

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This gentleman is an excellent teacher

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this was SO helpful for me to understand what PDEs were thank you

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It's amazing. Watching you solving problems, really makes these calculous so much fun. Please makes some videos on Fourier series, wave equation, heat equation problems. Many of us don’t understand the topics well.

[@azzteke]

calculus please

[@georgesadler7830]

Professor Crawford thank you for an awesome lecture on Partial Differentiation with powerful examples.

[@TomRocksMaths]

glad it was helpful!

[@swarnilpal3382]

Your explanation of this topic is great ! I appreciate your efforts

[@TomRocksMaths]

Thanks Swarnil!

[@jakebroster1720]

Finished my maths degree about 4 years ago and I absolutely love this explanation!

[@TomRocksMaths]

awesome - thanks :)

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My new favourite channel on KZbin. Superb content.

[@TomRocksMaths]

Thanks Jack

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Thank u so much!! After a devasting morning in which I didnt understand a thing of my calculus class, I come across your video and feel theres hope again!

[@TomRocksMaths]

glad it helped!

[@PreetiSingh-hh5xq]

Love the way you explain things. It's been a while since I had to do these problems (in college) - I love that these vids inspire me to crank out a few practice problems, just for the fun of it. Thanks Tom :)

[@TomRocksMaths]

You're very welcome Preeti :)

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Are you Indian ?

[@sleka_ke4577]

After I saw the tats, I came straight to the comments, Inspiring. Thanks

[@PanzerfaustBR]

The most important information in all of multivariate calculus is as follows: the symbol ∂ for the partial derivative is called "cursive d", but it's OK to spell it in LaTeXese as "\partial"! :D Great video! I simply love this part of Calculus, it is incredibly beautiful.

[@TomRocksMaths]

Agreed.

[@electozgaming7609]

Really nice explanation. Subbed

[@TheMusicbegins]

A good teacher, like a good entertainer first must hold his audience's attention, then he can teach his lesson. You are a good teacher as you can grab the attention of all students watching you.

[@TomRocksMaths]

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This video helps me a lot when answering my quizzes. Thanks 🤟

[@TomRocksMaths]

Happy to help!

[@ashafaghi]

Excellent, as always

[@larrykent196]

I do like your approach to explaining the subject. Easy to follow great video and thank you for sharing. Cheers!

[@TomRocksMaths]

thanks Larry!

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Thank you so much! This was very useful!

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Even foto learning math but english this Is Avery good video. Thanks Tom. Please continue doing great videos!!!

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FAB TUTORIAL PROFESSOR TOM WILL LOOK FOWARD TO VIEWING MORE

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Wow this video is great! I just got through DE and I was curios what came after. You are really awesome about the way you break it down and explain it.

[@TomRocksMaths]

Glad it was helpful!

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Best explanation I have ever seen ❤

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Amazing!Well explained.

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just doing this at year 1 undergrad at leicester and it makes more sense now, thanks

[@TomRocksMaths]

Happy to help :)

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A fantastic math teacher ! The student's are so lucky to have you. I just finished high school and to be honest I'm not good at math's at all. I can't blame the teacher's, but it would've been more fun and more easier to learn math's with teacher like you.

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Thanks for the lecture! Super clear.

[@TomRocksMaths]

Glad it was helpful Anthony!

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made it so simple to understand!

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Best Doctor of Mathematics

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This was amazing. Thank you so much for this video!

[@TomRocksMaths]

happy to help!

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Not even close to being first ;_;

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Better luck next time papa...

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Man, if I had access to this video when I was taking high school Physics, my life would have been so much easier! Sadly, the Internet wouldn't be prevalent in homes for another 5-10 years (I'm 50 now, and when I needed it would have been around age 18, or a nice clean 2^5 years ago), so I wouldn't have had a proper grasp on Schrödinger's equation for a while yet. ... No, we weren't learning about Schrödinger's equation yet, but an essay that I did for Physics involved it and I didn't quite understand what was going on. This would have helped a lot!

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Excellent review very clear

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Amazing video and professor!

[@TomRocksMaths]

Glad you liked it!

[@diyanbalavar7851]

Hiiii Tom. Just wanted to thank you❤😍 I really liked that and it was useful a lot. A presentation with clarity😊

[@TomRocksMaths]

Glad it was helpful!

[@MarkoPetejan]

OMG! For the first time it is so clear to me. That graph, that writing I suddenly understand it all. I implied there were some kind of rules for the example, I suppose you can lookup for, so that was not a problem neither. And then, just like that: WTF is second derivative?! How can I picture this in my mind? And could not continue past 10 minute mark. But subscribed to have a chance of more of those 10 minute revelations.

[@davidvose2475]

Very well done. Easy to follow.

[@TomRocksMaths]

Glad it was helpful!

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Its just amazing the way you explained

[@TomRocksMaths]

Awesome, glad you enjoyed it :)

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This is so well explained!

[@TomRocksMaths]

glad it was helpful!

[@aarentanwheilung9384]

Fantastic explanation

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You are a nice teacher. You explain math in such a way that I think everyone can understand and that is great : )

[@TomRocksMaths]

Thank you! 😃

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I just watched the first 6:50min and it cleared up a lot.

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Wow, so glad to have stumbled on your channels it's Amazing thanks ,🙏

[@TomRocksMaths]

Welcome!

[@NateCrownwell]

awesome! You are an amazing maths teacher!

[@TomRocksMaths]

Thanks Nate :)

[@sabahhadhood603]

Thanks so much. In my view, the lecture is very easy for this subject and difficult examples must be solved to understand the partial derivative, such as, how to find out d^3y/dx^3 of f(x,y)=0.

[@TomRocksMaths]

I explain how to solve simple PDEs like the one you mention here: kzbin.info/www/bejne/q6vXm6uofL5pgNk

[@albinbergstrand3180]

Cool Im on my 2nd year of studying electrical engineering and i haven't heard of the partial differentiation before, only partial integration. But this wasn't tricky at all nice teaching!

[@TomRocksMaths]

Glad you enjoyed it Albin :)

[@jorgelechon8044]

Thanks for another video. I can't understand very well English. But your explanation is awesome.

[@TomRocksMaths]

Awesome, thanks Jorge!

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cleared up my misunderstandings in this topic, thx prof❤️

[@TomRocksMaths]

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I'd like to understand that better, so I'm going to search for it.

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Super explained , thank you

[@TomRocksMaths]

Glad you enjoyed it Paja :)

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Great explanation!

[@TomRocksMaths]

Thanks Sam!

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Dang.. oxford is lucky to have a hot doctor and mathematician

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Great explanation in only around 18 minutes!

[@TomRocksMaths]

Glad you liked it

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This guy KNOWS Maths 😣♥️

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Crystal clear explanation sir. I am from India.❤️ On 21.01.2021

[@TomRocksMaths]

glad it helped!

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Takes me back to C2, A level maths

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Love the way you explain it

[@TomRocksMaths]

glad it was helpful!

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Idk how I got here I don’t even do maths but he’s lowkey legendary anyways so I’m not complaining

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[@TomRocksMaths]

Thanks Muhin!