Let two variables x and y are related to a given implicit relation f(x,y) =0. In general, this defines y as a multi-valued function of x. However, the simplest way of dealing with such a function is to introduce a third variable z as under:
Z = f(x, y)
However, z = 0 for all x and y related by f(x,y) =0. Thus the given implicit function can be studied by relating those values of x and y which makes z equal to zero in a explicit function z = f(x,y).
As x and y vary in any way, independently or not, the complete differential of z is given by
Dz = fx dx + fy dy
If, (x,y) are values showing z =0 and if, dx and dy are variations from these values keeping z = 0, then z = 0.
Thus, fx dx + fy dy = 0
Hence, dy/dx = fx/fy