in the "i know its true, but what does it mean" category, i've never quite understood why the derivative of the area of a circle is the circumference.

d(pi*r^2) = 2*pi*r is mathematically obvious (and, in fact, the 1 meter gap is a side effect of that very relationship, i'm sure, though i'm too rusty to prove it right now).

but still, its always been a mathematical fact and convenience; it never gelled in my head as to exactly *why* it should be that way.