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Math Help - nth Derivative in Proof by Induction

  1. #1
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    Arrow nth Derivative in Proof by Induction

    Hey, I've been working through some proof by induction problems, and most of them are manageable, but then I saw one which involved differentiation which I have no idea what to do. Could someone please lead me in the right direction?

     \displaystyle \mbox{Prove that the nth derivative of } y=ln(x), x>0 \mbox{ is: }<br />
\frac{d^ny}{dx^n} = \frac{(-1)^{n+1}(n-1)!}{x^n}

    I have the basic steps done:

    ===================

    Step 1: Prove for n = 1


    \dfrac{dy}{dx} [ln(x)] = \dfrac{1}{x}

    --------

    \dfrac{d^1y}{dx^1} = \dfrac{(-1)^2(1-1)!}{x^1}

    \dfrac{d^1y}{dx^1} = \dfrac{1}{x} = \dfrac{dy}{dx} [ln(x)] \mbox{ \therefore True }

    ===================

    Step 2: Assume for n = k

    \dfrac{d^ky}{dx^k} = \dfrac{(-1)^{k+1}(k-1)!}{x^k}

    ===================

    Step 3: Prove for n = k+1

    I know how to start this, but how do I continue?

    Thanks in advance for help. BG
    Last edited by BG5965; February 13th 2011 at 03:52 AM. Reason: Clarification
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  2. #2
    Senior Member
    Joined
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    For

    <br />
n=k+1<br />

    please calculate

    <br />
\dfrac{d}{dx} \;  \dfrac{d^ky}{dx^k} <br />
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