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In lecture #2.6 Professor Leonard, in his proof of the chain rule ends up doing something like this:

dy   du
-- * --
du   dx

He then cancels the 'du' top and bottom to complete his proof. But the 'dy/dx' form of indicating a derivative is surely not really a fraction is it? If we use the Newtonian notation there is no 'fraction' at all. The chain rule certainly works and you can see that the answer is the same as if you just distribute a polynomial raised to whatever power, however the above 'cancellation' seems very strange to me. It it were a fraction we'd end up with 'something over something' but we don't -- there is no fraction so how can canceling work? He just glosses over this so I hafta ask you guys.

Ray Andrews
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    That "cancel" method is not a proof, as you noticed. So ... ask Professor Leonard. Perhaps you missed some of what he said at that point? Or perhaps he made a mistake? Those of us here cannot tell which. – GEdgar Oct 30 '19 at 11:43
  • I'm watching videos on Toob, I can't ask him. – Ray Andrews Oct 30 '19 at 15:21

2 Answers2

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Your questions has been answered already.

Check the following links for more.

https://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

Fede1
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    Excellent, lots of meat on that link, thanks. – Ray Andrews Oct 30 '19 at 15:31
  • Sheesh. IS! ISN'T! IS! ISN'T! I'd propose that the infinitesimal is like zero in that it never has any size, but unlike zero in that it can form real ratios with other infinitesimals that have been multiplied or divided by real numbers. Thus dy/dx is a real ratio even tho both dy and dx have no size. Isn't it just that simple? – Ray Andrews Oct 30 '19 at 16:24
  • If some points have not been covered fully, I advise you to look more in depth the meaning of differential. In fact, this can answer your question. Moreover, make sure you fully understand the meaning of infinitesimals.

    Overall, explanation is kinda correct although mathematically quite "painful" to read haha.

     – Fede1 Oct 30 '19 at 18:06
  • Well, much to learn that's for sure. Anyway it's interesting that this stuff is still being debated. – Ray Andrews Oct 30 '19 at 21:33
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These notations like $\frac{du}{dx}$ are just represented in fractions to ease calculations. Even in ur case u used fraction division to cancel out du. This notation just represent derivatives . We can't cancel out them like fractions so we just made their notation in a way such that one can use fractional rules to solve.

Who am I
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  • I didn't use fractions, the professor did and I don't like it, it seems very fishy. And you can't have it both ways: if it isn't a fraction then you can't use fractional rules, canya? You're trying to say there is no cake but we can eat it anyway. – Ray Andrews Oct 30 '19 at 15:33
  • If there was no cake only empty space then it would be hard for DUMB people to eat. So we made empty space a cake so it is easy to visualise and to eat. – Who am I Oct 30 '19 at 17:27
  • Yeah, I'm beginning to see that I should back off on this question, there's a bit of faith involved, at least for now. – Ray Andrews Oct 30 '19 at 21:35