Question

If `Z=x^2 tan-1y /x-y^2 tan -1 x/y del` 

Prove that `(del^z z)/(del_ydel_x)=(x^2-y^2)/(x^2+y^2)`

Answer 1

`Z=x^2 tan-1y/x-y^2 tan -1 x/y` 

Diff.W.r.t.y partially, 

`del_z/del_x=x^2 x^2/(x^2+y^2)xx -y/x^2 +tan^-1  y/x.2x-y^2 y^2/(x^2+y^2)xx1/y` 

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

Diff. w.r.t y partially , 

`(del^2z)/(del_ydel_x)=-x^2[-y. 2y/(x^2+y^2)^2+1/(x^2+y^2)]+2 x^2/(x^2+y^2)-[-y^3. (2y)/(x^2+y^2)^2+(3y^2)/(x^2+y^2)]`

=`[(2y^3x^2)/(x^2+y^2)^z+(-x^2)/(x^2+y^2)]+2 x^2/(x^2+y^2 )+2y^4/((x^2+y^2)^2)-(3y^2)/(x^2+y^2)` 

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

=`(x^2-y^2)/(x^2+y^2)` 

∴`(del^2  Z)/(del_y del_x)=(x^2-y^2)/(x^2+y^2)`

Hence proved.