Expert: Paul Klarreich - 2/1/2012

Question
Mr. Klarreich,

I have a question regarding one of your past answers, at http://en.allexperts.com/q/Calculus-2063/difference-quotient.htm

The original question is:

Evaluate the difference quotient for the given function.  Simplify your answer.

F(x)= (1/x)

I'm in the beginning of my first semester of Pre-Calculus.  We are going over an introduction to functions.  According to my instructor, we are only using differential quotients right now to give us a small taste of Calculus.  We haven't actually used differential quotients for anything yet; we are just plugging functions into the formula.  My textbook uses this formula for differential quotients:

f(x+h)-f(x)
-----------
    h

I fully understand how to get to this point:

  x  - (x + h)
----------------
   x(x + h)
--------------- =
      h

The next step, per your previous answer, should bring me here:


    - h
------------- =
  hx(x + h)

Can you explain what I should be doing to get between these two steps?

Thanks for your time.

Answer
Questioner:Matt
Country:Nevada, United States
Category:Calculus
Private:No
Subject:Pre-Calculus I

Question:

Mr. Klarreich,

I have a question regarding one of your past answers, at http://en.allexperts.com/q/Calculus-2063/difference-quotient.htm

The original question is:

Evaluate the difference quotient for the given function.  Simplify your answer.

F(x)= (1/x)

I'm in the beginning of my first semester of Pre-Calculus.  We are going over an introduction to functions.  According to my instructor, we are only using differential quotients right now to give us a small taste of Calculus.

>>>>  And how, exactly, does it taste?

We haven't actually used differential quotients

>>>>  DIFFERENCE QUOTIENTS, please

for anything yet; we are just plugging functions into the formula.  My textbook uses this formula for differential quotients:

f(x+h)-f(x)   
-----------
    h

>>>>>>>>>>>>>>>

OR:

f(x2) - f(x1)     <<< a difference of two things
-------------     <<< a quotient of two things.
 x2 - x1         <<< a difference of two things



I fully understand how to get to this point:

  x  - (x + h)
----------------
   x(x + h)
--------------- =
      h

The next step, per your previous answer, should bring me here:


    - h
------------- =
  hx(x + h)

Can you explain what I should be doing to get between these two steps?

Thanks for your time.
..........................................................
Start with:

  1       1
------ -  ---
x + h      x
-----------------
      h

Clear fractions;  multiply each term by the LCD:  x(x + h)

x(x + h) 1     x(x + h)1
----------- -  ---------
 x + h          x
----------------------------
      h x(x + h)


Cancel a few things:

x....... 1     ..(x + h)1
----------- -  ---------
 .......          ..
----------------------------
      h x(x + h)

  x - (x + h)
-----------------
  h x(x + h)


  x - x - h      <<<< remove parentheses
-----------------
  h x(x + h)

   - h      <<<< simplify
---------------
  h x(x + h)

and you are on your way.