When to Use the Chain and Quotient Rules

Date: 06/12/98 at 23:35:42
From: Matthew
Subject: Order of differentiation

Hi, I'm doing a maths subject called "Introductory Calculus" and 
learning about the various forms of differentiation. I've come across 
such questions as:

   Differentiate:

             (2x + 1)^2
      f(x) = ----------
               2x + 4 

In situations like this, do I use the quotient rule first or the chain 
rule first?

Date: 06/16/98 at 23:06:37
From: Doctor Mateo
Subject: Re: Order of differentiation

Hello Matthew,

In the specific example that you give, you begin by recognizing that 
the function is a rational function - it looks like a fraction with a 
variable in the denominator (the bottom). 

Because you are looking for the first derivative of a rational 
function, you begin by applying the quotient rule first. As you apply 
the quotient rule, you encounter a term in the numerator with a power.  
So what happens here is that you end up applying the chain rule in the 
process of doing the quotient rule.
 
Consider the two functions below:

              (4x - 5)^3                            ( (4x - 5) )^3 
   (I) g(x) = -----------      and     (II) h(x) =  (----------)
               (5 + 3x)                             ( (5 + 3x) ) 

In I) above, the exponent is part of the term in the numerator (top).  
The exponent does not go with the term in the denominator.

In II) above, the exponent is on the outside of the entire fraction, 
which means that it actually belongs to the numerator (top) and the
denominator (bottom) because:
  
          ( (4x - 5) )^3    (4x - 5)^3
   h(x) = (----------)    = ----------
          ( (5 + 3x) )      (5 + 3x)^3 


In II) you would apply the chain rule first, since the exponent goes 
with the numerator and the denominator. Then you would do the quotient 
rule on the rational function inside. Alternatively, you could put the 
exponent with the numerator and the denominator then apply the 
quotient rule first, since you now have a rational function, but then 
you would have to use the chain rule twice. 

In I) you would apply the quotient rule first since you have 
a rational function, and then do the chain rule on the part that 
involves the exponent. 

I hope this helps you distinguish the order a little better. Have fun 
differentiating!

-Doctor Mateo,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/