Expert: Paul Klarreich - 10/12/2009

Question
Find the derivative of 8/√(x-2) using the definition of a derivative.

I have attempted this problem but for some odd reason I can't seem to reach the answer.

WORK:
lim    8/√(x+h-2) - 8/√(x-2)
h->0              h

After this point I don't know what algebra to use to get the answer, I tried conjugate, common denominator, squaring the whole thing.

Answer
Questioner: Davis
Country: United States
Category: Calculus
Private: No
Subject: Finding derivative using the definition of a derivative
Question: Find the derivative of 8/√(x-2) using the definition of a derivative.

I have attempted this problem but for some odd reason I can't seem to reach the answer.

WORK:

lim    8/√(x+h-2) - 8/√(x-2)
h->0   ---------------------
             h

After this point I don't know what algebra to use to get the answer, I tried conjugate, common denominator, squaring the whole thing.
..................................................
Why don't we start with some lowest common denominator stuff:

LCD = √(x+h-2) √(x-2)

as in let's multiply by it, to clear fractions.  After every term gets 'hit' by it:

your original limit:

lim    8/√(x+h-2) - 8/√(x-2)
h->0   ---------------------  
             h
becomes:

lim     8√(x-2) - 8√(x+h-2)
h->0   ---------------------  =
        h √(x+h-2) √(x-2)


lim      √(x-2) - √(x+h-2)
h->0   8 ------------------
        h √(x+h-2) √(x-2)

NOW rationalize (multiply by conjugate) and we take out that '8'.

The conjugate is (√(x-2) + √(x+h-2))

lim       √(x-2) - √(x+h-2)    (√(x-2) + √(x+h-2))
h->0   8 --------------------  --------------------
        h √(x+h-2) √(x-2)     (√(x-2) + √(x+h-2))

lim             (x-2) - (x+h-2)   
h->0   8 -------------------------------------
        h √(x+h-2) √(x-2) (√(x-2) + √(x+h-2))

lim             x - 2 - x - h + 2   
h->0   8 -------------------------------------
        h √(x+h-2) √(x-2) (√(x-2) + √(x+h-2))

lim                - h
h->0   8 -------------------------------------
        h √(x+h-2) √(x-2) (√(x-2) + √(x+h-2))

Things will finish up nicely now -- I'll leave that to you.  Isn't it great when things fall into place?
[Note: These special symbols cause trouble.  I hope the square root symbol does not get messed up.]