Find Fxx of the following equation?

Question

f(x, y) = 5x arctan(x/y)
I found Fx of this but I have trouble finding Fxx

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Answer 1 · 2018-05-24T01:57:30

$(10y^3)/(x^2+y^2)^2$

$f(x,y) = 5x tan^-1(x/y) implies$

$f_x(x,y) = 5tan^-1(x/y)+5xtimes (1)/(1+(x/y)^2)times 1/y$
$qquad = 5tan^-1(x/y)+5 (xy)/(x^2+y^2) implies$

$f_{x x}(x,y) = 5times (1)/(1+(x/y)^2)times 1/y$
$qquad qquad qquad qquad +5((x^2+y^2)times y-xytimes 2x)/(x^2+y^2)^2$

$qquad qquad = (5y)/(x^2+y^2)+(5y(y^2-x^2))/(x^2+y^2)^2$
$qquad qquad = (5y{(x^2+y^2)+(y^2-x^2)})/(x^2+y^2)^2$
$qquad qquad = (10y^3)/(x^2+y^2)^2$