Proof of the Chain Rule

[tjkubo]

I've been using this site to help me understand the basics of calculus. So far I'm okay with most of it. But I just don't understand the proof of the chain rule they use. Can anyone explain it for me, or else give me a different proof?

Here is the link:
http://www.themathpage.com/aCalc/chain.htm

 

[galactus]

I have a small problem with their last line. \(\displaystyle {\Delta}g\)

approaching 0 is undefined. It's in the denominator. For instance:

\(\displaystyle \lim_{x\to\0}\frac{1}{x}=undefined\). A no-no!.

This proof looks rather watered down.

Why prove the chain rule?. Most have a bad enough time using it, let

alone proving it. Unless your curious and want to understand, then

a proof of the chain rule can be found online and in many calc texts.