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Math Help - Proof help!

  1. #1
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    Proof help!

    I'm having a lot of problems figuring out this question. If someone could at least point me in the right direction that would be great.

    "Prove that for every positive integer n , (d^n / dx^n)[xe^-x] = (-1)^n e^-x(x-n)"
    Last edited by mr fantastic; December 7th 2008 at 06:22 PM. Reason: Question restored by Mr F (the OP deleted it for 'privacy' reasons - there ain't no such thing on the web, honey).
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  2. #2
    MHF Contributor

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    Do this by induction.
    \frac{{d^k }}{{dx^k }}\left[ {xe^{ - x} } \right] = \left( { - 1} \right)^k e^{ - x} \left( {x - k} \right)
    \begin{array}{lr} {\frac{{d^{k + 1} }}<br />
{{dx^{k + 1} }}\left[ {\left( { - 1} \right)^k e^{ - x} \left( {x - k} \right)} \right]} & { = \left( { - 1} \right)^k \left[ { - e^{ - x} \left( {x - k} \right) + e^{ - x} } \right]} \\<br />
{} & {=\left( { - 1} \right)^{k + 1} \left( {e^{ - x} } \right)\left[ {x - \left( {k + 1} \right)} \right]} \\<br />
\end{array}
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