Expert: Paul Klarreich - 10/2/2006

Question
Find an algebraic expression for the difference quotient (f(1+deltX)- f(1))/ deltX when f(x)= x^2-(1/x). Simplify the expression as much as possible. Then determine what happens as deltX approaches 0.

Answer
David Asks in Category Calculus ...
 
Subject:  Difference quotient
 
Question:  Find an algebraic expression for the difference quotient (f(1+deltX)- f(1))/ deltX when f(x)= x^2-(1/x). Simplify the expression as much as possible. Then determine what happens as deltX approaches 0.
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Hi, David,

You're going to do a lot of this, and it's a bit messy, but essential.

Here's a way to set it up:

WARNING: USE A FIXED FONT, LIKE COURIER, TO VIEW THIS.

Write your function:
             1
f(x) = x^2 - ---
             x  

Write the function expressions used in the DQ:(I'm going to write dx to save typing.)  

When you substitute your  "1 + dx" or something like it, BE SURE TO PARENTHESIZE CORRECTLY.  Nothing is more important than that, and failure to do that leads to most of the difficulties in this computation.
                           1
f(1 + dx) = (1 + dx)^2 - -------  and simplify a bit:
                        1 + dx

                               1
f(1 + dx) = 1 + 2dx + dx^2 - -------
                            1 + dx

               1
f(1) = (1)^2 - --- = 1 - 1 = 0
               1

Ok, now.  DIFFERENCE QUOTIENT means quotient of differences, so you subtract and then divide.


f(1 + dx) - f(1) =

                  1
1 + 2dx + dx^2 - ------ - (0) =
                1 + dx
                  1
1 + 2dx + dx^2 - ------
                1 + dx

Now divide:  Be careful in dividing a fraction.  To divide by dx, multiply by 1/dx.

f(1 + dx) - f(1)
---------------- =
      dx

1 + 2dx + dx^2     1     1
-------------- - ------ ---
     dx         1 + dx  dx

Do some algebra, like combining fractions over the LCD of  (1 + dx)dx

(1 + 2dx + dx^2)(1 + dx) - 1
----------------------------
       (1 + dx)dx


1 + 2dx + dx^2 + dx + 2dx^2 + dx^3 - 1
--------------------------------------
        (1 + dx)dx          


3dx + 3dx^2 + dx^3
------------------
    (1 + dx)dx   

Now a factor of dx cancels.  THIS IS THE VITAL PART.  IF IT DOESN'T CANCEL HERE, YOU SCREWED UP THE ALGEBRA.

      
3 + 3dx + dx^2
---------------
  (1 + dx)

Now that's your difference quotient.  The fun part comes next.  You let  dx -> 0.  [Really, all you do is put zero for dx.]  Later you will write this as:

lim   [THE DIFFERENCE QUOTIENT]
dx->0


3 + 3dx + dx^2
--------------- -->
  (1 + dx)

3 + 3(0) + (0)^2
---------------- = 3
  (1 + 0)

That's it.