The Chain Rule using Leibniz notation

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Description:
The chain rule can also be presented in Leibniz notation. This is useful when we have a bunch of funky variables and want to be explicit about which is which, and also shows the intuition behind the formula a little bit more clearly.
Learning Objectives:
1) Compute the derivative of a composition using Leibniz notation
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!
Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.
This video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.
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Опубликовано:

2 окт 2024

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Комментарии : 21

@darshanp8224 3 года назад

Wow I asked same doubt to my mam that "How can we cancel a notation" she din't say anything about this. Well thanks for your information sir.

@2002budokan 3 года назад

There is an error in this video that is not mathematically significant, but is very important in respect to an important name in the history of mathematics. The German mathematician's name is Leibniz, not Liebniz. This error is probably due to the fact that the name Newton is always written before Leibniz. Whereas Calculus is something they both find at the same time, unaware of each other. Perhaps when we talk about Calculus in the next century, we should change the order of its inventors to Leibniz & Newton.

@drewscampfire 2 года назад

I love this! I feel like I have an intuitive understanding of the chain rule now.

@1951split 4 года назад

It's LEIbniz, not Liebniz. ;) Lipe (like "wipe") Nits (like "bits")

@joy4467 8 месяцев назад

iblis

@punditgi Год назад

Lye-bnits, please. The name is spelled Leibniz in German. Otherwise, an excellent lecture! Danke schoen! 😃

@RemotHuman 10 дней назад

kind of confusing because in the top formula f(x) means just the outer function but in the bottom formula f means the composition of the outer and the inner function - at least I think so

@alexvanmeter497 5 лет назад

Been trying to understand this for weeks!! now it makes clear sense. Thank you

@sayanjitb 3 года назад

Dear sir, I understand the chain rule clearly. But my question is why do we write the factor of two derivatives in the chain rule (e.g. here)? I know it mathematically but how can you explain it in the light of intuition and graphical representation? TIA

@adaptercrash 2 года назад

Waste of money if you ask me I am in 1200 AD before this chain thing I simply just didn't like it. You don't even really need it and all this stuff you could get anyway.

@tiagoantonio2343 2 года назад

that's a brillant and beautiful class, thank you

@joshuasanders644 5 лет назад

thanks!

@alexgian9313 2 года назад

Respectfully, I have to disagree. When I was first introduced to modern notation (Spivak) which is probably closest to Lagrange, I also thought that Leibniz was clearer, but this was probably due to the cumulative effects of poor teaching methods and to habits. Inttroducing intermediate variables merely muddies the waters and also leads to some horrible ambiguities that can bite you later. In the Lagrange notation you use, (f(g(x)))' is simply (f o g)'(x) So we have (f o g)'(x) = f'(g(x)) . g'(x) Which is simple enough, until you realise that even the "x" is redundant, getting: (f o g)' = f'(g) . g' or in Spivak notation: D(f o g) = (D f)g . (D g) What could be clearer or simpler than that? And no more spurious ambiguous confusion of the type df/dx=df/dy . dy/dx Never mind the creepy implicit cancellation that spooks first-time students. That stuff belongs in museums. Since you're dealing with univariate functions, why bother about the name of the "wrt" variable, anyway? sin'(x)=cos(x), sin'(bumf)=cos(bumf), etc... Hell, sin'=cos , nice and simple if you view it functionally.

@tncreations1267 10 месяцев назад

Here, you are saying that you also thought that Leibniz's notation was clearer when you were first introduced to modern notation. As a beginner myself, I could easily understand what is actually happening in the chain rule when it is presented in Leibniz notation rather than Lagrange's. I have tried to understand Lagrange's at first but I found it bit confusing and it took me some time to get it. Now I can easily understand the chain rule in whatever notation it is presented because I was able to grasp the idea using Leibniz notation. Which I think is the point of this video.

@alexgian9313 10 месяцев назад

@@tncreations1267 Well, yes, inititially I did think that Leibniz was "clearer" but that was because I was more willing to accept the hand-waving and the "cancellation" of the dy. It worked, but it didn't help me understand.

@tncreations1267 10 месяцев назад

@@alexgian9313 I guess we are different types of learners then. 😀

@alexgian9313 9 месяцев назад

@@tncreations1267 All types of learning are good and to be applauded! I'd add that I'm not dismissing Leibniz notation out of hand. For many reasons, including that at some loftier interpretation level, dy might indeed cancel! It's just that I'd much prefer the functional notation to be taught first. It would solve a lot of problems that mess with beginner students' minds, including the ambiguity of y as a coordinate or y as a function.