Quotient Function Mnemonic

Date: 03/11/2002 at 20:23:16
From: Ryan DeMont
Subject: Grade 12 Calculus

I get really turned around when trying to factor or expand derivatives 
of quotient functions.

Help me to differentiate the following:

x/((x^2)-4)

Date: 03/17/2002 at 16:18:24
From: Doctor Nbrooke
Subject: Re: Grade 12 Calculus

Good evening, Ryan, and thanks for writing to Dr. Math.

This is a very easy mistake to make, and one that I made quite a few 
times myself when I was in Calculus. Remember that the derivative of 
the quotient function has the form 

   (f/g)' = [g*f' - f*g']/[g^2]

However, it is much easier to remember 

   LOW DEE HIGH OVER HIGH DEE LOW OVER LOW SQUARED 

where LOW is the denonominator of the quotient and HIGH is the 
numerator. So we'll organize our functions, using your example:

LOW   x^2 - 4   DEE LOW  2x
HIGH  x         DEE HIGH 1

So we get 

(LOW DEE HIGH - HIGH DEE LOW)/(LOW SQUARED) 
= ((x^2-4) - x*2x) /(x^2-4)^2 
= (x^2 - 4 - 2x^2) / (x^2-4)^2
= (-x^2 - 4) / (x^2 - 4)^2

And there we go.  I hope this mnemonic device works for you. If you 
have any more questions, please feel free to write back.

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