With kind permission from Richard Sincovec:

"No, you can't use the second derivative on line segments. There's an explanation in linear algebra, and a possibly simpler one in topology once you reach the point where you can understand all the 3-ball and 2-sphere stuff, but the easiest explanation might be a sort of real-world analogy. AutoCAD works in 3-D, and most of us probably had our first real experience with derivatives in physics anyway, with stuff like v = xdt and so forth, so that approach might be the best.

Let's imagine a polyline as being the path a ball has travelled. As you remember from physics, a derivative is a rate of change. So, since we're assuming the polyline represents the position of the ball over time, the first derivative would be the velocity the ball is travelling at any particular instant in time. For straight segments, this velocity vector is in the same direction as the direction of travel. For curves, this is in a line tangent to the curve at that point. AutoCAD returns this direction vector, or first derivative, as simply a coordinate, the endpoint of a vector; the vector is assumed to start at (0,0,0). And, niftily enough, it looks like AutoCAD defined things so that the length of the vector is the length of the line for line segments, and the radius of the curve for arcs. (I wouldn't recommend using this function to get lengths or radii, though... It might have just turned out that way in the few items I checked in some poking around...)

Now, let's look at the second derivative. We remember from physics that this is acceleration, or how much the ball's velocity is changing. In other words, the second derivative is another directional vector, this one showing which way some invisible force would need to push the ball to make it travel along our polyline. And, again from our physics, this directional acceleration vector would be in the plane of the curve, and perpendicular to the direction of travel. In AutoCAD terms, the direction of this vector is perpendicular to the curve, toward its radius. And again, because of the way the math works, the length of the direction vector is equal to the radius of the curve.

However, with lines, the direction of travel is not changing. In physics terms, the lateral acceleration is 0. This implies the second derivative of a line segment must ALWAYS be 0, regardless of its direction. (Conversely, the second derivative of a curve must ALWAYS be non-zero.)"