If `x^xy^yz^z = c.` Then show that at `x = y = z;` `(del^2z)/(delx\ dely) = -(x log ex)^(-1)`

If x^(x)y^(y)z^(z)=c. Then show that x=y=z(del^(2)z)/(del x backslash del y)=-(x log ex)^(-1)

If f (x,y) =e ^(xy) then (del^(2)f)/(delx dely) at (1,1) is:

If v(x,y,z) = x^(3) + y^(3) + z^(3) + 3xyz , show that (del^(2)v)/(del y del z) = (del^(2)v)/(del z del y)

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

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

If z (x + y) = x ^ (2) + y ^ (2) show that [(del z) / (del x) - (del z) / (del y)] ^ (2) = 4 [1 - (del z) / (del x) - (del z) / (del y)]

If u=(x^(2)+y^(2)+z^(2))^(-1/2) show that x(del u)/(del x)+y(del u)/(del y)+z(del u)/(del z)=-u

If u=(x^(2)+y^(2)+z^(2))^(-1/2) show that x(del u)/(del x)+y(del u)/(del y)+z(del u)/(del z)=-u

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