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Question Number 27784 by abdo imad last updated on 14/Jan/18

let give  f(x) = e^(−(1/x))    with f(0)=0  1) is f derivable in point 0?  2)prove that  f^((n)) = F_n (x) e^(−(1/x))   with F_(n ) is rational function  3) calculate   f^((6))  (x) and  f^((9)) (x)  .

$${let}\:{give}\:\:{f}\left({x}\right)\:=\:{e}^{−\frac{\mathrm{1}}{{x}}} \:\:\:{with}\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{is}\:{f}\:{derivable}\:{in}\:{point}\:\mathrm{0}? \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:{f}^{\left({n}\right)} =\:{F}_{{n}} \left({x}\right)\:{e}^{−\frac{\mathrm{1}}{{x}}} \:\:{with}\:{F}_{{n}\:} {is}\:{rational}\:{function} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\:{f}^{\left(\mathrm{6}\right)} \:\left({x}\right)\:{and}\:\:{f}^{\left(\mathrm{9}\right)} \left({x}\right)\:\:. \\ $$

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