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Math Help - Induction

  1. #1
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    Induction

    Prove by induction the nth derivative of e^(a*x) is (a^n)*e^(a*x) for all n that exist in the natural numbers greater than or equal to 1.

    That was the conjecture I made from the taking a few derivatives.

    After showing p(1) is true and assuming p(k) is true, how do I start proving p(k+1) is true.

    I thought about starting with p(k) and multiplying by a but I don't think that will work in this situation since the derivative must be taking.
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  2. #2
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    Hi

    You assume that p(k) is true which means \frac{d^k e^{ax}}{dx^k} = a^k e^{ax}

    You want to show that p(k+1) is true : \frac{d^{k+1} e^{ax}}{dx^{k+1}} = a^{k+1} e^{ax}

    Just take the derivative of \frac{d^k e^{ax}}{dx^k} = a^k e^{ax}
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