Consider the real function f(x,y)=xy(x(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}+y^{2})^{-N},in the respective cases N = 2,1, and 1/2. Show that in each case the function is differentiable (C^{[tex]\omega[/tex]}) with respect to x, for any fixed y-value.

whats the strategy for proving C^{[tex]\omega[/tex]}-differentiability here? i have to show with induction that f is still continuos after n+1 derivatives with respect to x, or am i wrong?

seems like, one has to look after a pattern in the continued derivatives of f, for write down the (n+1)nth derivative.

on the other hand, one might have the differential quotient in mind, taking the limit, meeting a boundary condition(in the complex domain the radius of convergence), but i could only think of the 1th derivative here.

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# Surface differentiability

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