Expert: Paul Klarreich - 12/30/2007

Question
I need to use the definition of the derivative to find the derivative of f(x)=1/(sqr rt x+1).  I think my problem is in the algebra.  Thanks

Answer
Questioner:   Mike
Category:  Calculus
Private:  No
 
Subject:  Derivatives
Question:  I need to use the definition of the derivative to find the derivative of f(x)=1/(sqr rt x+1).  I think my problem is in the algebra.  Thanks
.......................................................
Hi, Mike,

I think your problem lies in a bit of carelessness, too.  Did you mean to write:

    1
-----------
sqrt(x + 1)

or

    1
-----------
sqrt(x) + 1

You actually wrote the second one.  The first would be written 1/sqrt(x+1)

So I will do the second one, since that is what you wrote.
                  1
f(x + h) = ----------------
          sqrt(x + h) + 1

               1
f(x)    = ------------
          sqrt(x) + 1

                     1                 1      
f(x+h) - f(x) = ---------------  - ------------
               sqrt(x + h) + 1    sqrt(x) + 1


                sqrt(x) + 1 - (sqrt(x + h) + 1)
f(x+h) - f(x) = ---------------------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1)


                sqrt(x) + 1 - sqrt(x + h) - 1
f(x+h) - f(x) = ------------------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1)

                sqrt(x) - sqrt(x + h)
f(x+h) - f(x) = ------------------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1)
Rationalize:


                sqrt(x) - sqrt(x + h)         sqrt(x) plus sqrt(x + h)
f(x+h) - f(x) = ------------------------------ --------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1) sqrt(x) plus sqrt(x + h)



                x - (x + h)
f(x+h) - f(x) = --------------------------------------------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1)(sqrt(x) plus sqrt(x + h))



                x - x - h
f(x+h) - f(x) = --------------------------------------------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1)(sqrt(x) plus sqrt(x + h))


                   - h
f(x+h) - f(x) = --------------------------------------------------------
               (sqrt(x + h) + 1)(sqrt(x) + 1)(sqrt(x) plus sqrt(x + h))

Divide by h:

f(x+h) - f(x)                    - 1
-------------- = --------------------------------------------------------
     h          (sqrt(x + h) + 1)(sqrt(x) + 1)(sqrt(x) plus sqrt(x + h))


Limitize (h -> 0) [OK,OK, so I made up that word.]


   f(x+h) - f(x)                    - 1
lim -------------- = --------------------------------------------------------
        h          (sqrt(x + 0) + 1)(sqrt(x) + 1)(sqrt(x) plus sqrt(x + 0))


                   - 1
= ---------------------------------------------------
  (sqrt(x) + 1)(sqrt(x) + 1)(sqrt(x) plus sqrt(x))

         - 1
= ----------------------------
  (sqrt(x) + 1)^2(2 sqrt(x))


         - 1
= ----------------------------
 2 sqrt(x) (sqrt(x) + 1)^2

Check using derivative rules:


u = sqrt(x) + 1, du/dx = 1/2sqrt(x)

f(u) = u^-1,  f'(u) = - u^-2

The rest is easy.