i need to prove that this equation is true(adsbygoogle = window.adsbygoogle || []).push({});

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\frac{d^n}{dx^n}\left(x^{n-1}e^{1/x}\right) = (-1)^n\frac{e^{1/x}}{x^{n+1}}

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i tried by induction

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f_n(x) \equiv x^{n-1}e^{1/x}

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g_n(x) \equiv (-1)^n\frac{e^{1/x}}{x^{n+1}}

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this is the pattern if i keep differentiating the base case.

[itex]

\frac{{d^{k + 1} }}{{dx^{k + 1} }}f_{n + 1} (x) = x\frac{{d^{k + 1} }}{{dx^{k + 1} }}f_n (x) + (k + 1)\frac{{d^k }}{{dx^k }}f_n (x)

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then i tried to differentiate the "n" equation inorder to get to the n+1 equation

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\frac{{d^{n + 1} }}{{dx^{n + 1} }}f_n (x) = \frac{{d^{} }}{{dx^{} }}g_n (x) = ( - 1)^n {\rm{[}}e^{1/x} ( - \frac{1}{{x^2 }}{\rm{)}}x^{n + 1} + (n + 1)x^n e^{1/x} {\rm{] }} \\

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but it didnt work

??

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# Homework Help: How to prove this equation

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