What Is an Understandable Proof of the Chain Rule?

[zeion]

Homework Statement

I'm looking for a proof for the chain rule that is relatively easy to understand. Can someone show / link me one? Thanks.

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Google is your friend:

http://math.rice.edu/~cjd/chainrule.pdf

 

[hunt_mat]

I posted my calculus notes with a basic (non-rigouous) derivation of the chain rule in, you could have a look at those.

 

[zeion]

For the pfd file, I don't understand the middle of page 2 where it says

"..and use the second equation applied to the right-hand-side with k =
[g'(x) + v]h.. Note that using this quantity for k tells us
that k -> 0 as h -> 0, and so w -> 0 as h -> 0."

How did they choose that substitution for k?

 

[zeion]

anyone?

 

[spyrustheviru]

how about this one?
let u(x) be a differentiable fuction in [a,b] with values in [a',b'] and y=f(x) a differentiable fuction in [a',b'].
Let Δx be a randomly picked difference x2-x1. That causes a change Δu on u(x), while Δu causes a change on y=f(u).
We have Δu=(u'(x)+n1)*Δx. You can easily verify by looking at the graph that the line connection the points (x1,u(x1)) and (x2,u(x2)) has a slope equal to the value of the derivative of u on x1 plus a number n1 to compensate for the fact that Δx isn't zero(and thus this line isn't the tanget on x1).
The same applys to Δy=((f'(u)+n2)*Δu.
When Δx-->0 , n1,n2-->0
Δx/Δy=(f;(u)+n2)*(u'(x)+n1)
We calculate the limit of the fraction when Δx-->0 and it is equal to f'(u)*u'(x)=(dy/du)*(du/dx)
system has gone crazy and won't show the math symbols, sorry for the formating

It's the proof from Louis Brand's book "Advanced Calculus", paragraph 52- The chain rule