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 x2x1. 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 wont show the math symbols, sorry for the formating
It's the proof from Louis Brand's book "Advanced Calculus", paragraph 52 The chain rule
