The tangent plane to the surface S at the point 
is
the plane containing both tangent lines of the x and y cross-sections.
Suppose f has continuous partial derivatives. Those tangent lines are given
by
and
An equation of the tangent plane to the surface z=f(x,y) at the point
is
The function
is called the linearization of f at 
.
If z=f(x,y), then f is differentiable at (a,b) if 
can
be expressed in the form
where 
and 
as 
.
For a function z=f(x,y), we define the differentials dx and dy to be independent variables; that is, they can be given any values. Then the differential dz, the total differential, is defined by
Differentials give us approximate changes to a function in the neighborhood of a point, generally when dx and dy are small.
If the partial derivatives 
and 
exist near (a,b) and are
continuous at (a,b), then f is differentiable at (a,b).
This is a simple generalization of the tangent line, of course: in each cross-section, the tangent plane contains the tangent line. One of the interesting results is that differentiability in just two directions means that the function is differentiable (i.e., smooth).