Question

Find Fxx of the following equation?

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

Answer 1

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

Explanation:

`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`