this question from optimaization theory course please solve it with steps and details

this is the definition of Gateux differentiable

Let f:R2R be a mapping defined by f((x,y))={x2+y2x3xy30if(x,y)=(0,0)if(x,y)=(0,0) Let v=(v1,v2)tR2\{(0,c)t}, prove the existence of Dv(f(0,0)t). and prove f is Gatux-differentiahte on Rn. Definitione.4 "Gateaux d.fferentiability" Let be an open set of Rn, ixe Say That fis Gateaux ol.fferentioble at a if (1) vRn,Dvf(a) exists (*) The mapping df(a,)v=RnRmdf(a,v)Qvf(a) Let f:R2R be a mapping defined by f((x,y))={x2+y2x3xy30if(x,y)=(0,0)if(x,y)=(0,0) Let v=(v1,v2)tR2\{(0,c)t}, prove the existence of Dv(f(0,0)t). and prove f is Gatux-differentiahte on Rn. Definitione.4 "Gateaux d.fferentiability" Let be an open set of Rn, ixe Say That fis Gateaux ol.fferentioble at a if (1) vRn,Dvf(a) exists (*) The mapping df(a,)v=RnRmdf(a,v)Qvf(a)