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Finding Derivative Using 'Definition of Derivative'

  1. Sep 26, 2012 #1
    1. The problem statement, all variables and given/known data
    mv3l1y.jpg


    2. Relevant equations
    As seen above in question.


    3. The attempt at a solution
    Well, I substituted f(x) into the definition of derivative equation and then multiplied the expression by its conjugate, thus getting the expression in the following form:
    f'(a) = lim (a + h)2/3 - a2/3 / h [(a + h)1/3 + a1/3]
    ..........h→0

    Now, from this point on, it's just algebra and I am having trouble manipulating this expression so that I can divide the "h" from the denominator. I know I am supposed to use the "hint" given in the question but how would I incorporate that into finding the answer?


    Thanks.
     
  2. jcsd
  3. Sep 26, 2012 #2

    SammyS

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    Well, multiplying by the conjugate, doesn't work here because getting a difference of squares isn't helpful when working with the cube root.

    Use the hint given.

    Let [itex]a=\sqrt[3]{x+h}\,,[/itex] and [itex]b=\sqrt[3]{x}\ .[/itex]
     
  4. Sep 26, 2012 #3
    I did that but it ended up getting to this expression after some simplifications:
    [ (x + h)2/3(x)1/3 - (x + h)2/3(x)1/3 ] / h
    as h → 0.

    You can already see what the problem is here. The numerator results in 0, which is subsequently divided by h.
     
  5. Sep 27, 2012 #4

    SammyS

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    So, what did you multiply by ?

    You want to multiply [itex]\sqrt[3]{x+h}\,-\,\sqrt[3]{x}[/itex] by something that results in cubing each of those terms. Right?
     
    Last edited: Sep 27, 2012
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