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Limit definition of derivative problem

  1. Feb 22, 2013 #1
    1. The problem statement, all variables and given/known data

    Using the definition of derivative find f'(x) for f(x) = x - sqrt(x)

    2. Relevant equations

    None.

    3. The attempt at a solution

    lim h --> 0 : ((x + h) - sqrt(x + h) - x + sqrt(x))/h

    1 - (sqrt(x + h) - sqrt(x))/h

    Multiply by conjugate..

    1 - h/(h*(sqrt(x) + sqrt(x+h)))

    1 - 1/(sqrt(x+h) + sqrt(x))

    lim as h --> 0 makes it: 1 - 1/2sqrt(x)

    -------------------------------------------------------
    QUESTION:

    My question is, is there a way to solve this problem without multiplying by the conjugate? My friend says there's more ways but I don't see how?

    Also, how come using the limit definition of derivative with
    (f(x) - f(a)) / (x - a) yields zero?
     
  2. jcsd
  3. Feb 22, 2013 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    There are general rules for obtaining the derivative of powers like f(x) = x^k (k = any number, positive or negative).

    RE: your second question: the ratio you write does NOT give 0. After all, you just finished finding the result for f(x) = sqrt(x) and a = 0: you did not get zero then, did you?
     
  4. Feb 22, 2013 #3
    I got it, was making a silly error :biggrin:
     
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