Hi,
An implicit function in x and y is two dimensional, while z=f(x,y) is three dimensional.
In the two dimensional case, x and y are in one plane and that's it. In the three dim case the function produces a third dimension quantity that depends on both x and y.
Examples:
r^2=x^2+y^2 (this is implicit in x and y, R is a constant, more properly written R^2=x^2+y^2)
z=x^2+y^2 (this is three dimensional, z is a variable).
Implicit differentiation:
y^2=2y+x+2
2yy'=2y'+1
Differentiation on z=f(x,y):
z=x^2+y^3
dz/dx=2*x
dy/dx=3*y^2
Implicit functions allow the expression of multivalued functions without declaring two or more different functions.
For example
R^2=x^2+y^2
allows us to define an entire circle in one expression rather than having to define it in two expressions:
y=sqrt(R^2-x^2) (top half)
y=-sqrt(R^2-x^2) (bottom half)