Associated Topics || Dr. Math Home || Search Dr. Math

Differentiating a Quotient

```
Date: 08/29/97 at 00:39:06
From: KATHY GLEGHORN
Subject: Functions

Given that g(x) = x-4/x+3, find: {g (x+h) - g(x}/h)

```

```
Date: 08/29/97 at 08:33:16
From: Doctor Anthony
Subject: Re: Functions

This is really the way to prove the rule for differentiating a
quotient - i.e. by letting h -> 0 you will find g'(x).

(x+h)-4     x - 4
g(x+h) - g(x) = --------- - --------
(x+h)+3     x + 3

(x+3)[x+h-4] - (x-4)(x+h+3)
=  ----------------------------
(x+3)(x+h+3)

(x+3)(x-4) + h(x+3) - (x-4)(x+3) - h(x-4)
=  -------------------------------------------
(x+3)(x+h+3)

hx + 3h - hx + 4h
=   -------------------
(x+3)(x+h+3)

7h
=    ---------------
(x+3)(x+h+3)

And dividing by h we get

7
-------------
(x+3)(x+h+3)

If we let h -> 0 then we get

7
---------
(x+3)^2

Check this result by finding g'(x), using the usual quotient formula

(x+3) - (x-4)         7
g'(x) = -------------  =   -------
(x+3)^2        (x+3)^2

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search