z(x;y)=1/y^2+x^2-1
rarr dz=(delz)/(delx)dx+(delz)/(dely)dy
(delz)/(delx) is calculated as the derivative of z(x;y) by x assuming that y is constant.
(delz)/(delx)=cancel((d(1/y^2))/dx)+dx^2/dx-cancel((d(1))/dx)=2x
Same thing for (delz)/(dely):
(delz)/(dely)=(d(1/y^2))/dy+cancel(dx^2/dy)-cancel((d(1))/dy)=-2/y^3
Therefore: dz=2xdx-2/y^3dy