"f_x(x,y)=y^3\\cdot{x^2+y^2-2x^2 \\over (x^2+y^2)^2}=y^3{y^2-x^2 \\over (x^2+y^2)^2},\\ (x,y)\\not=(0,0)"
"f_y(x,y)=x\\cdot{3y^2(x^2+y^2)-2y^4\\over (x^2+y^2)^2}=xy^2{3x^2+y^2\\over (x^2+y^2)^2},\\ (x,y)\\not=(0,0)"
The partial derivatives are defined at (0, 0).
"f_x(0,0)=\\lim\\limits_{h\\to0}{1\\over h}(f(0+h,0)-f(0,0))=\\lim\\limits_{h\\to0}(0-0)=0"
"f_y(0,0)=\\lim\\limits_{h\\to0}{1\\over h}(f(0,0+h)-f(0,0))=\\lim\\limits_{h\\to0}(0-0)=0" Therefore
"f_x(0,0)=f_y(0,0)=0"
Let "x=rcos\\theta, y=r\\sin\\theta"
"\\lim\\limits_{(x,y)\\to(0,0)}f_x(x,y)=\\lim\\limits_{r\\to0}\\bigg[r^3\\sin^3\\theta{r^2\\sin^2\\theta-r^2\\cos^2\\theta \\over (r^2\\sin^2\\theta+r^2\\cos^2\\theta)^2}\\bigg]=""=\\lim\\limits_{r\\to0}\\big[r^3\\sin^3\\theta(-\\cos(2\\theta))\\big]=0"
"\\lim\\limits_{(x,y)\\to(0,0)}f_x(x,y)=0=f_x(0,0)"
"f_x(x,y)" is continuous at (0, 0)
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