Now, for the equation F(x,y,z)=0, the implicit function theorem states that under relatively mild conditions, there exists a function Z(x,y), so that F(x,y,Z(x,y))=0 holds IDENTICALLY in an open region about a solution of F(x,y,z)=0.
We may find the expression for dZ/dx and d(dZ/dx)/dx by differentiatiang our identity with respect to x.