If `z(x+y) = x^2 + y^2` show that `[(delz)/(delx) - (delz)/(dely)]^2 = 4[1-(delz)/(delx)-(delz)/(dely)]`

If f(x,y) = 1/(sqrt(x^2+y^2)) then show that x(delf)/(delx)+y(delf)/(dely) = -f

If u= x^3-3xy^2 , show that (del^2u)/(delx^2)+(del^2u)/(dely^2) =0

If u =sqrt(x ^(4) +y ^(4)) show that x (delu)/(delx) + y ( del u)/(dely) =2u.

If z(x+y)= x^2 +y^2 show that (frac{delz}{delx}-frac{del z}{del y})^2 = 4 (1- frac{delz}{delx}+frac{delz}{dely})

If u = x^(4) + y ^(3) + 3x ^(2) y^(2) + 3x ^(2) y find (del u)/(delx), (delu)/(dely), (del^(2) u)/(delx^(2)) , (del^(2)u)/(delx dely).

If u = (x^2+y^2+z^2)^(-1/2) then prove that (del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2) =0

If f = (x+y)/(sqrt(x-y)) then x (delf)/(delx) +y (delf)/(dely) is :

If u = cos ^(-1) ((x)/(y)) prove that x (del u)/(delx) + y (del u)/(dely)=0.

Let w(x, y, z)=(1)/(sqrt(x^(2)+y^(2)+z^(2)))(x, y, z) != (0, 0, 0) . Show that (del^(2)w)/(delx^(2))+(del^(2)w)/(dely^(2))+(del^(2)w)/(delz^(2))=0