Earlier we saw how the two partial derivatives  and  can be thought of as the slopes of traces.  We want to extend this idea out a little in this section.  The graph of a function  is a surface in  (three dimensional space) and so we can now start thinking of the plane that is “tangent” to the surface as a point.

Let’s start out with a point  and let’s let  represent the trace to  for the plane  (i.e. allowing x to vary with y held fixed) and we’ll let  represent the trace to  for the plane  (i.e. allowing y to vary with x held fixed).  Now, we know that  is the slope of the tangent line to the trace  and  is the slope of the tangent line to the trace .  So, let  be the tangent line to the trace  and let  be the tangent line to the trace .

The tangent plane will then be the plane that contains the two lines  and .  Geometrically this plane will serve the same purpose that a tangent line did in Calculus I.  A tangent line to a curve was a line that just touched the curve at that point and was “parallel” to the curve at the point in question.  Well tangent planes to a surface are planes that just touch the surface at the point and are “parallel” to the surface at the point.  Note that this gives us a point that is on the plane.  Since the tangent plane and the surface touch at  the following point will be on both the surface and the plane.

What we need to do now is determine the equation of the tangent plane.  We know that the general equation of a plane is given by,

where  is a point that is on the plane, which we know already.  Let’s rewrite this a little.  We’ll move the x terms and y terms to the other side and divide both sides by c.  Doing this gives,

Now, let’s rename the constants to simplify up the notation a little.  Let’s rename them as follows,

With this renaming the equation of the tangent plane becomes,

and we need to determine values for A and B. 

Let’s first think about what happens if we hold y fixed, i.e. if we assume that .  In this case the equation of the tangent plane becomes,

This is the equation of a line and this line must be tangent to the surface at  (since its part of the tangent plane).  In addition, this line assumes that  (i.e. fixed) and A is the slope of this line.  But if we think about it this is exactly that the tangent to  is, a line tangent to the surface at  assuming that .  In other words,

is the equation for   and we know that the slope of  is given by .  Therefore we have the following,

If we hold x fixed at  the equation of the tangent plane becomes,

However, by a similar argument to the one above we can see that this is nothing more than the equation for  and that it’s slope is B or .  So,

The equation of the tangent plane to the surface given by  at  is then,

Also, if we use the fact that  we can rewrite the equation of the tangent plane as,

We will see an easier derivation of this formula (actually a more general formula) in the next section so if you didn’t quite follow this argument hold off until then to see a better derivation.