(Expand this Subsection to read an explanation of the technical definition)

Here again, for your reading pleasure, is the  technical definition of differentiability: a function is differentiable at a point a if there's a linear transformation T such that

(The double bars in that equation mean ``length of a vector'.')  Let's think about what that equation means: the limit is zero, so as x approaches a, the numerator becomes much smaller than the denominator. The denominator is just the distance from x to a. Since T is linear, T(x-a) = T(x) - T(a), and the part of the numerator inside the bars can be written in a somewhat more illuminating way:

But wait...T is a tangent plane (or ``tangent 3-space'', etc) so f(a) must equal T(a), and in the above equation they'll cancel. So the numerator becomes even simpler—

—and we can rewrite the above limit as

If that limit is zero, then as x approaches a (along any path, of course) the numerator is much smaller than the denominator, which means that the linear transformation approximates f(x) very well near a. You might say that in a very small neighborhood of a, f(x) is ``just about'' linear. This is what we were trying to visualize earlier in this section.

This definition of differentiability is, unfortunately, not very practical. If we're interested in deciding whether a function is differentiable or not (and also in finding the particular linear transformation that approximates it), we have to use a test which will be described below.

Created by Mathematica  (November 6, 2004)