Chain Rule Notation

Date: 04/21/99 at 04:55:25
From: Eric 
Subject: Chain Rule

I'm trying to figure out these questions:

   Formula : (f*g)'(x)= g'(x) . f'[g(x)]

  1)  f(x) = 2x+6
      g(x) = 3x-4
   (f*g)'(x) = 3 . 2 = 6        I know that g'(x)= 3 but how about  
                                f'[g(x)]? How does 2 come about? I 
                                don't understand how it's done. 
  
  2)  g(x) = 2x^2 + 5
      h(x) = x^4 
    (g*h)'(x) = 4x^3 . 4x^4
              = 16x^7           It's the same here. I know how to 
                                differentiate h(x) but I got stuck on 
                                g'[h(x)]. How does 4x^4 come by? 

Please help me,
Thanks.

Date: 04/21/99 at 10:19:17
From: Doctor Mitteldorf
Subject: Re: Chain Rule

Dear Eric,

The chain rule can be taught in such a way that it's quite 
transparent, or it can be made utterly mysterious with bad notation.  
It looks as if you've been a victim of the latter. 

The chain rule is about taking the derivative of a function of a 
function. Instead of f being a function of x, we have f is a function 
of g, and g is a function of x. In this notation, the chain rule can 
be written:

     df/dx = df/dg  .  dg/dx

It seems almost obvious when you write it that way. Just "cancel out" 
the dg's in the numerator and denominator.

In your example (1),

 f(x) = 2x+6
 g(x) = 3x-4

The teacher gave you a notation that's deliberately confusing. You 
have to remember that the x in these equations is a dummy variable.  
The top equation just says 

   f is a function that takes its argument, multiplies it by 2, then 
   adds 6.  

The x is there just as a placeholder. You can replace it with a or b 
or theta or phi and the equation says exactly the same thing.

But in this case, you want to replace it with g:

f(g) = 2g+6  

Is it obvious why I want to replace the x by a g? It's because f*g 
means "f composed with g," or "the function f taken of the function 
g."  (Using the * for "composed with" can be a confusing notation too 
if you are mixing it in an equation in which * means simply 
multiplication. You have wisely used a dot for multiplication.)

Coming back now to problem 1, let's do it two ways. First, we'll 
actually find f*g and differentiate it. Second, we'll use the chain 
rule. Then we'll be in a position to check that the two answers are 
the same.

First,

 f(g) = 2g+6
 g(x) = 3x-4

Substituting the second equation into the first, you have

  f(x) = 2(3x-4)+6 = 6x-2

It's obvious, then, that f'(x) = 6.

Second, we'll use the chain rule:

df/dx = df/dg  .  dg/dx
      =   2    .    3    = 6

Hence, we get the same answer both ways.      

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/