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

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

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[ProfessorLeonard]

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

[mewsicman]

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.

[mothman]

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