(adsbygoogle = window.adsbygoogle || []).push({}); Chain Rule - intuitive "Proof"

Suppose y = f(u), and u = g(x), then dy/dx = dy/du * du/dx.

In an intuitive "proof" of the chain rule, it has this step: dy/dx = [tex]\lim_{\Delta x \to 0} \frac {\Delta y}{\Delta x}[/tex] = [tex]\lim_{\Delta x \to 0} \frac {\Delta y}{\Delta u}[/tex] * [tex]\frac {\Delta u}{\Delta x}[/tex]

My question is, why multiply by [tex]\frac {\Delta u}{\Delta u}[/tex]? I know mathematically, it's because [tex]\frac {\Delta u}{\Delta u}[/tex] = 1, and multiplying by 1 doesn't change the function, but I'm looking for a philosophical reason. I found a quote by the user mathwonk in an old thread, which says: "it seems plausible that the best linear approximation to a composite function, is obtained by composing the best approximations to the component functions. On the other hand for a linear function, composing means simply multiplying." Can someone expand on this?...

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# Chain Rule - intuitive Proof

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