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Definition of the derivative to find the derivative of x^(1/3)

  1. Mar 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Use the definition of the derivative to find the derivative of x^(1/3)

    2. Relevant equations



    3. The attempt at a solution

    [(x+h)^(1/3) - x^(1/3)]/h

    I do not know where to go from here. If it were a square root I could conjugate.
     
  2. jcsd
  3. Mar 8, 2012 #2

    SammyS

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    Hello Martinc31415. Welcome ton PF !

    The difference of cubes can be factored, [itex]a^3-b^3=(a-b)(a^2+ab+b^2)\,.[/itex]

    So, suppose you have the difference of cube roots, [itex]\displaystyle P^{1/3}-Q^{1/3}[/itex]. In this case, [itex]\displaystyle P^{1/3} = a\,\ \text{ and }\ Q^{1/3} = b\,.[/itex]

    Multiplying [itex]\displaystyle \left(P^{1/3}-Q^{1/3}\right)[/itex] by [itex]\displaystyle \left(P^{2/3}+P^{1/3}Q^{1/3}+Q^{2/3}\right)[/itex] will give [itex]\displaystyle \left(P^{1/3}\right)^3-\left(Q^{1/3}\right)^3=P-Q\,.[/itex]

    Thus, [itex]\displaystyle \left(P^{2/3}+P^{1/3}Q^{1/3}+Q^{2/3}\right)[/itex] acts as the "conjugate" for [itex]\displaystyle \left(P^{1/3}-Q^{1/3}\right)\,.[/itex]
     
  4. Mar 8, 2012 #3
    oohh...

    I would not have thought of that, ever!

    Thanks a ton.
     
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