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Prove the Power Rule (calculus)

  1. Sep 11, 2005 #1
    Ok guys, I'm new here and I need some help with a math problem...

    The problem asks me to prove the power rule ---> d/dx[x^n] = nx^(n-1) for the case in which n is a ratioinal number...

    the one stipulation is that I have to prove it using this method: write y=x^(p/q) in the form y^q = x^p and differentiate implicitly...assume that p and q are integers, where q>0.

    Thanks for any help

    Matt
     
  2. jcsd
  3. Sep 11, 2005 #2

    Fermat

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    Are you allowed to assume that "d/dx[x^n] = nx^(n-1)" is true for n = an integer?

    Have you differentiated "y^q = x^p" implicitly yet ?
     
  4. Sep 11, 2005 #3
    yes I have and I came up with....


    q(y^(q-1))(dy/dx) = p(x^(p-1))

    but I'm stuck after I get here...i guess I could isolate dy/dx, but I'm not sure where to go from there...any help?
     
  5. Sep 11, 2005 #4

    Hurkyl

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    Well, do you know what y is?
     
  6. Sep 11, 2005 #5

    Fermat

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    if y = x^(p/q)
    then
    y' = (p/q)x^(p/q - 1)

    Use the expression you got for for the implicit differentiatoin and the expression y^q = x^p and manipulatre them to end up with the required result, shown above.
     
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