# verify that (f)xy=(f)yx. if f(x,y)=sin^-1(y/x)

### 1 Answer | Add Yours

`f(x,y)=sin^(-1)(y/x)`

`f_x=1/sqrt(1-(y/x)^2)(-y/x^2)`

`=x/sqrt(x^2-y^2)(-y/x^2)`

`=-(y/x)(x^2-y^2)^(-1/2)`

`f_(xy)=-(1/x){(x^2-y^2)^(-1/2)+y(-1/2)(x^2-y^2)^(-3/2)(-2y)}`

`=-(1/x)1/(x^2-y^2)^(3/2)(x^2-y^2+y^2)`

`=-x/(x^2-y^2)^(3/2)` (i)

`f_y=1/sqrt(1-(y/x)^2)(1/x)`

`=(x^2-y^2)^(-1/2)`

`f_(yx)=(-1/2)(x^2-y^2)^(-3/2)(2x)`

`=-x/(x^2-y^2)^(3/2)` (ii)

From (i) and (ii), we have

`f_(xy)=f_(yx)`