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Math Help - i tried to prove this equation in differential way..

  1. #1
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    i tried to prove this equation in differential way..

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    You are trying to prove by induction that \frac{d^n}{dx^n}\bigl(x^{n-1}e^{1/x}\bigr) = (-1)^n\frac{e^{1/x}}{x^{n+1}}. So you assume the result for n=k, and you are trying to prove it for n=k+1 by differentiating one more time. Where you are going wrong is that you are finding the (k+1)th derivative of the same function. But the result for n=k+1 refers to the (k+1)th derivative of the function x^{k}e^{1/x} = x\bigl(x^{k-1}e^{1/x}\bigr). You can differentiate that k+1 times by regarding it as a product (as I just indicated) and using Leibniz's rule together with the inductive hypothesis.
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    x^{k}e^{1/x} = x\bigl(x^{k-1}e^{1/x}\bigr)

    so i made the k+1 derivative and i need to multiply both sides by the n+1 derivative of x
    in order to make it the k+1 expression
    ??
    so i need to do the derivative of x*f(x)=1*f(x)+x*f'(x)

    of the left side

    i have both f(x) and f'(x)
    so if i will substitute
    it will work??
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