Calculus 1 Lecture 2.6: Discussion of the Chain Rule for Derivatives of Functions

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For comment about cancellation of "du," here is what I hope you will find to be an adequate response: I suppose it does suffice for a Calc 1 student as the COMPLETE proof takes some time and as soon as epsilon is put on the white-board, eyes start to gloss over... BUT, in a simple proof using Leibniz notation (as this is), this actually DOES represent a fraction as we would think of it and DOES "Cancel-Out," but only BEFORE taking the limit! So, if you were to use the definition of dy/du*du/dx AS A LIMIT, the "du's" would cancel (this works only if Delta u and Delta x BOTH approach Zero, please read the last few sentences). The reason why I tell the students to think of it that way is because in the proof, it is Delta 'u' that "Cancels." It absolutely IS DIFFERENT FROM du, because "du" would contain with it the idea of a limit, but that is where if comes from. You would define Delta 'u' to be the change in g(x), as x changes, and Delta 'y' to be the change in f(u) as g(x), which is 'u', changes. Because we are finding the derivative of f(x), and not f(u), that 'u' acts as a dummy variable. Its is introduced in Leibniz notation as a fraction. If you were to do the definition of a derivative, the one with all the limits, you would see that the numerator would contain "change in f(x)" and the denominator would contain "change in x" One way to develop the Chain Rule (derivative of a composition) is actually to use fractions, as we know them, INTO that definition. Notation-wise, we get dy/du*du/dx (AFTER taking the limits). Now, the reason why I say "One way" and "Simple proof" is because that proof is not complete and only proves one case where Delta U approaches zero when Delta X approaches zero. To deal with the case where is DOESN'T, you need the more involved proof that more people are familiar with. But thank you very much for the comment and I hope this clears up any ambiguity. P. Leonard

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37:37 I love how keep reminding what calculus is about. I was once a student who don't lost track of what derivative actually is when I was in my freshmen days, that's why I've never understand calculus before. But now that I'm re-learning it, everything is starting to make sense because of how amazing professor Leonard's teaching skill is. I'm really grateful for him.

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I'm new to Calculus, and I'm studying it by myself, at the age of 31. If only this had been my very first lesson on the Chain Rule, it would have spared me a few weeks of frustration and self-doubt. I get it now. This is such a great resource! Thank you so much!

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Hello Professor Leonard. My name is Luis. I took calculus back in high school and I scored so poorly on the college board exams. My teacher taught me well but I failed to understand some parts. My teacher did explain to me what these rules and theorem mean. He was just like you. He said the steps matter the answer doesn't. As I recall. I'm going to take Calculus 1 at a university. I'm taking notes and I am watching your videos 100% all the way to the end. Your videos have become a great help so far. Thank you for your videos. I know you aren't getting paid. I am also a math major. As an adult I understand this more versus when I was in high school. Thanks you. Unfortunately, you won't reply haha because you haven't looked back at your videos but I express my gratitude to you. Thank you!!!!!

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[@Goldenblade14]

I'm glad that you cover both methods of the Chain Rule. I have a crappy teacher who has only taught the chain rule through u-substitution, completing neglecting the now significantly easier derivative of f(g(x)). Even then, you explained u-subbing far better than she did. You did me a big service, and I'm sure I'm far from the only you did this for.

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These videos are so easy to digest that I actually watched beyond what my calc professor has taught us so far, and I didn't even notice it. I checked my notes and my professor stopped at sum and difference derivatives and I'm already on the chain rule. I feel like that's truly the mark of a great teacher when you don't feel so overwhelmed you have to stop at one or two lessons.

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WOW!!! I finally understand the chain rule! Thanks Professor!

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So like others, I am taking an online class, but I just want to thank you as well... you're amazing. "follow the dy/dx". Makes SO much more sense. Thanks you so much for making these videos.

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It blew my mind when he asked “what about dy/dx” @ 8:00 Nobody had ever explained chain rule to me like that. Man is a miracle worker

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1:14:01 I had a hard time seeing this. I had to take one more step in order to clarify it for myself that the derivative of x^3 is not going to be multiplied into the 3x^2. I did: d/dx[x^3] + d/dx[csc(x^3)] and broke those two things up first so I could see that one derivative didnt have anything to do with the other derivative. Im glad someone in class asked so that you could clarify that one. Very helpful videos. Thanks Prof. Leonard

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Thank you so much. My professor spent like 20 minutes on this and I was so confused.

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I finally get chain rule.. I must have read the stupid text book a dozen times and watched the class lesson we are given at least three times.. Made zero sense as to what exactly I was doing.... Now I get it! Wow thanks man!

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