Chain of Reasoning
(Justification of the Chain Rule)

Consider some function y = f(g(x)), the composition of two differentiable functions.

If we write u = g(x), then y = f(____).

Suppose now that x is changed by a small amount, x. This will cause u to change by an amount ____. If this last quantity is not zero, then:

y
------
x 		 
=
  		  
------
   		 
.
  		  
------
  
Taking the limit of both sides as x --> 0 (and applying one of the Limit Laws) we get the following:
lim
 
x--> 0 		y
------
x 		 
=
  		lim
 
x--> 0 		    
--------
      		 
.
  		lim
 
x--> 0 		     
--------
     

It is easy to see that as x-->0, u-->______. Why? Because we know that u = g(x), so that u = g(__________) - g(______).

Hence if x approaches 0, we have:

lim
x--> 0 		u
  		=
  		lim
x--> 0 		________________
  		= 0 since g(x) is ________, hence ________.
So, the limit becomes:
dy
------
dx 		 
=
  		lim
 
x--> 0 		y
------
x 		 
=
  		lim
 
u--> 0 		     
--------
      		 
.
  		lim
 
x--> 0 		     
--------
     
Which is really the product of the derivatives:
dy
------
dx 		 
=
  		d  
----------
d     		 
.
  		d  
----------
d    
There is one flaw: u might equal zero; but we will not worry about this subtlety here.
Copyleft notice: Copyright © 1997, by Lana Golembeski -- "Share and Enjoy".
Non-profit educational use explicitly allowed and encouraged. May not use for any other purpose without written permission