Appearance of Formal Derivative in Algebra When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are coprime. My question is, should we be surprised to see the formal derivative here? 
Is there some way we can make sense of the appearance of the derivative (which to me is an analytic object) in algebra? I suspect it might have something to do with the fact that the derivative is linear and satisfies the product rule, which makes it a useful object to consider. It would also be interesting to hear an explanation which explains this in the context of algebraic geometry.
Thanks!

The formal derivative in a polynomial ring is just one example of a more general concept called a derivation. More explicitly, if $R, S$ are rings with $S$ an $R$ algebra, and $d : R \rightarrow S$ is a map which satisfies:
1) $d(a + b) = d(a) + d(b)$
2) $d(ab) = a d(b) + b d(a)$
Then $d$ is called a derivation. That formal derivative in a polynomial ring is just one example of a derivation, but there are many derivations that pop up all over the place in algebra and geometry.
In the general context of algebra, there is a big theory called differential galois theory, which studies differential field extensions (extensions of fields which have a derivation, such as $\mathbb{C}(x)$ for example).
In differential geometry, a smooth manifold has a derivation on its set of real-valued smooth functions $C^\infty(M)$, which is basically the usual derivative. The target of this derivation is not the set of functions, but the set of differential 1-forms $\Lambda^1(M)$.
These are just a couple of example, but derivations are important because of their algebraic properties, and because derivations show up naturally in many branches of mathematics. I can't speak for Algebraic Geometry, but I believe derivations play an important role in that theory as well.
