You know that trick where you invent some number ε such that ε2 = 0 and use it to, basically, take derivatives?
For example, (x+ε)2 = x2 + 2xε, so if we change x by some small amount ε then we change x2 by 2x times that amount. Thus the derivative of x2 must be 2x.
It turns out that trick has a name; it's calculation in the algebra of dual numbers, which I discovered by randomly poking around Wikipedia, and it's apparently used in at least some computer algebra systems to do differentiation. I didn't know that.
Edit (Monday, 4:36 pm): Charles of Rigorous Trivialities has pointed out that dual numbers are used extensively in deformation theory.