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I've seen a couple of proofs for the chain rule, and I know this probably sounds stupid, but I'm wondering why it can't be proved as follows:

given the real valued functions [itex]y=f(u), u=g(x) [/itex]

since [itex]dy, du, dx, [/itex] are all real valued functions as well

can't you just state:

[tex]\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} [/tex]

by the properties of real numbers?

can someone explain why this isn't an acceptable proof?

Also since I'm on the subject of differentials, does anyone know of any good books on the thoeries of differentials, because I've spent a lot of time thinking about this concept, and it seems to have a different meaning for different applications. I've heard that there are plenty of theories that explain what a differential is, and explains more about it's uses ( beyond the scope of a Calc 1-3 book). Any info would be greatly appreciated.

Thanks in advance!

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# Chain Rule / Differentials

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