Question

Considere a função definida por f(x,y)=x.$e^{x.y^2}$
a) Calcule as derivadas parciais $F_{x}$ (x,y) e $F_{y}$ (x,y) num ponto P (x,y)

Answer 1

$F_x(x,y)=(x)'e^{xy^2} +x(e^{xy^2})'\\ \\ F_x(x,y)=1.e^{xy^2}+xe^{xy^2}((x)'y^2+x(y^2)') \\ \\ F_x(x,y)=e^{xy^2}+xe^{xy^2}(1.y^2+x2yy') \\ \\ F_x(x,y)=e^{xy^2}+e^{xy^2}(xy^2+2x^2yy') \\ \\ F_x(x,y)=e^{xy^2}(1+xy^2+2x^2yy')$

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$F_y(x,y)=(x)'e^{xy^2} +x(e^{xy^2})' \\ \\ F_y(x,y)=0.e^{xy^2} +xe^{xy^2}.((x)'y^2+x(y^2)') \\ \\ F_y(x,y)=0 +xe^{xy^2}.(0.y^2+x2y) \\ \\ F_y(x,y)=xe^{xy^2}.(2xy) \\ \\ F_y(x,y)=2x^2ye^{xy^2}$