Expert: Paul Klarreich - 8/13/2011

Question
Find d[1/sqrt(x)]

So I know dy = F'(x) times dx.

I am just having a hard time finding the derivative,

I start out at:

[1/sqrt(x+dx)] - [1/sqrt(x)]  because that is f(x+dx) - f(x) and then convert both fractions so that they have the same denominator and I can subtract it out, but I can not seem to come up with the right answer, which is,

[(-1/2)x^-(3/2)]dx

Any help is appreciated,

Aaron

Answer
Questioner:Aaron
Country:Florida, United States
Category:Calculus
Private:No
Subject:Finding the differential of Y



Question:
Find d[1/sqrt(x)]

So I know dy = F'(x) times dx.

I am just having a hard time finding the derivative,

I start out at:

[1/sqrt(x+dx)] - [1/sqrt(x)]  because that is f(x+dx) - f(x) and then convert both fractions so that they have the same denominator and I can subtract it out, but I can not seem to come up with the right answer, which is,

[(-1/2)x^-(3/2)]dx

Any help is appreciated,

Aaron
..............................................
   1          1
----------- - --------
sqrt(x+dx)    sqrt(x)

Combine:

sqrt(x) - sqrt(x+dx)
---------------------
sqrt(x+dx) sqrt(x)


Rationalize:

sqrt(x) - sqrt(x+dx) sqrt(x) + sqrt(x+dx)
-------------------- --------------------
sqrt(x+dx) sqrt(x)   sqrt(x) + sqrt(x+dx)

         x - (x +dx)
------------------------------------------
sqrt(x+dx) sqrt(x) [  sqrt(x) + sqrt(x+dx) ]

         -  dx
------------------------------------------
sqrt(x+dx) sqrt(x) [  sqrt(x) + sqrt(x+dx) ]

Now divide by dx. (we put it back later, OK?)

         -  1
------------------------------------------
sqrt(x+dx) sqrt(x) [  sqrt(x) + sqrt(x+dx) ]

Finally  let  dx --> 0  (I promised to put it back later, OK?)

         -  1
------------------------------------------
sqrt(x) sqrt(x) [  sqrt(x) + sqrt(x) ]

        -  1
-------------------------
x [2sqrt(x) + sqrt(x)]


Simplify, then multiply by dx to get your differential.