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Math Help - Reciprocal Rule to verify the Power Rule

  1. #1
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    Reciprocal Rule to verify the Power Rule

    I'm stuck on this one:

    Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,

    \frac{d}{dx}(x^{-n}) = -nx^{-n-1}

    for all positive integers n.
    ----------------------------------------
    Playing this out, I did this:

    \frac{d}{dx}(x^{-n}) = -nx^{-n-1} = -nx^{-(n+1)} = (-n)(\frac{1}{x^{n+1}})

    Next, I tried to restate the function like this:

    x^{-n} = \frac {1}{x^n}

    and then I apply the Reciprocal Rule:
    \frac{d}{dx}[\frac{1}{x^n}] = \frac{(x^n)'}{(x^n)^2} = \frac{nx^{n-1}}{x^{2n}}

    None of this is seems to be leading me anywhere helpful. Can anybody see where I might be going wrong or suggest another tack?

    Thanks.
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  2. #2
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    You forgot a minus sign.

    \displaystyle \dfrac{d}{dx} \left( \dfrac{1}{x^n} \right) = - \dfrac{(x^n)^{'}}{(x^n)^2}=- \dfrac{nx^{n-1}}{x^{2n}}=-nx^{n-1-2n}=-nx^{-n-1}=-nx^{-(n+1)}=-n \, \dfrac{1}{x^{n+1}}
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  3. #3
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    Thanks. I forgot that the quotient rule requires a negative in front of the fraction. That solves the whole thing.
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