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### Quotient Function Mnemonic

```
Date: 03/11/2002 at 20:23:16
From: Ryan DeMont

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

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/
```
Associated Topics:
High School Calculus

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