You say that the du doesn't actually cancel but you never explain what actually happens (no one ever does). I find that this is required for me, personally, to get an intuitive knowledge of this subject (especially the notation). Can you explain intuitively, then rigorously, what happens with the du? I promise that I will be capable of following it.
polyopulis Like you, I have never managed to get an explanation of that from anyone. They say it's wrong but it works. Drives me crazy! I don't think they understand it either. Meanwhile, it has been six years since you wrote the comment I am replying to -- did you ever get a satisfactory answer from anyone?
Let's talk about notation and "notation". What is "3"? "3" is a bona-fide notation for "((1 + 1) + 1)", because "3" is a drop-in replacement for "((1 + 1) + 1)". They can be used interchangeably. What is "dx"? That is, what is the mathematical object or procedure for which "dx" is the drop-in replacement? I don't know. So, is "dx" really notation or "notation"? What about "dy/du * du/dx"? In dy/du, "u" is being used as a variable. In du/dx, "u" is being used as a function. So, why should they be treated the same? Newton had his notation, Leibniz had his "notation", Euler had his notation, and over the centuries, probably many other folks developed their own notation for calculus, as well. However, everyone who learns calculus has to come to terms with the Leibniz "notation". It's far and away the most popular. It's been said that using the Leibniz "notation" one can do a lot of calculus without ever understanding what in the world is really going on. To me, it's more of a pseudo-notation, or a notation contrivance, or a notational aid, and for that, it works great. You might consider it to be more of a helpful mnemonic than a mathematically precise notation. A mathematically rigorous calculus textbook might not even display the formula: (1) dy/dx = dy/du * du/dx. Instead, it might prove the Chain Rule as: (2) (f o g) ' (x) = f ' [g(x)] * g ' (x), where f and g are functions, and the "o" in "f o g" stands for "composite", as in the composite function (f(g))(x). (1) is more intuitive and useful in day-to-day operations after you learn the rigorous (2).
Wow. My online class uses openstax and the explanation of Leibniz Notation on there had my head spinning. This video is a complete lifesaver. Consider me a subscriber! Thanks!
@anacondaerslayer You take the derivative of "the whole thing" and get 2(3x+1), thats correct. Then you take that and multiply by the derivative of the inside and get: 2*(3x+1)*3 You just left out the x in "(3 + 1)", but other than that you have the right idea!
uggghhh i've been stuck so long on this fucking problem......... Y= u^3 - 5(u^3 - 7u)^2 , u = 1/x , x=4 please someone for the love of god help me!!!!!
The reference to fraction multiplication is confusing, and seems to imply that multiplication would be represented by the same notation. If y = (3x+1)(2x+2) and you applied the same logic to finding its derivative, wouldn't you also write dy/dx = dy/du * du/dx? Maybe once I see you explain the product rule, I will feel differently about the logic presented here. But right now, the info I have about Leibniz notation implies it might not actually "work." I know it works. I know we haven't been using broken notation for four centuries. :D
At about two minutes, he says that the du’s cancelling out like fractions is the wrong reasoning. Does anybody know any resources that would explain the right reasoning for that in detail?
Leibniz notation is pretty bad actually, it generally causes a lot of confusion towards new calculus students. But it's a standard, since Leibniz invented it. That's why everyone should learn this.
I cannot tell you how much trouble the abbreviated form of Leibniz notation, omitting the infinitesimal in the denominator, has caused me. I have never heard anyone explain what that is supposed to mean. Finally now I think I've got it figured out, about 50 years too late.
Amazing video!! Thanks a lot!! I had a really hard time understanding the usage of Leibniz' notation and especially in Implicit Differentiation but you made me understand a bit more!! :)
thank you so much, i like Leibniz notation so much better.. I think it's much more clear than prime notation and with this video it cleared up using the chain rule with Leibniz notation for me so much!! you da man!
Bello sono passati 10 anni, vedendo il nome assumerò che tu sia italiano. io avrei preferito svolgere prima il quadrato del binomio y=(3x+1)=9x^2+6x+1 e poi y'=18x+6 lui invece ha fatto un gioco ha assunto due funzioni una chiamata u=3x+1 e poi ha fatto u'=3 poi y=u^2 e poi y'=2u sempre per la powers rule. inseguito ha fatto così y'*u'=2(3x+1)*3 se fai i conti 6(3x)+6(1) dunque 18x+6
@ 1:53 "Even though I have to warn you the reasoning there is actually incorrect ..." Derivatives behave like fractions because fractions are exactly what they are. In fact that's the reason for Leibniz notation in the first place. The differentials dy & dx are very, very close to zero but neither of them is exactly zero. That's why you can write dy/dx as a fraction because it actually is a fraction (rational expression). It's the whole reason the chain rule works.
