More the first than the second. There are things other than the integers for which it makes sense to talk about "the primes". One example is: polynomials (with

More the first than the second. There are things other than the integers for which it makes sense to talk about "the primes". One example is: polynomials (with coefficients in, let's say, the complex numbers). In this case it turns out that the "primes" are exactly the linear polynomials (ax+b) where a is nonzero. There's a bit of ambiguity there, just as there is in the integers; 7 and -7 are "the same prime number", and x+3 and 5x+15 are "the same prime polynomial"; if we're going to say D(p)=1 then we need to pick which "version" of p has this property, and the obvious choice is the one of the form (x+a). So, now, if we apply the same definition as for integers to polynomials with these conventions, it says: (1) D(x+a) = 1 and (2) D(fg) = fD(g) + D(f)g when f,g are polynomials. And that turns out to give the exact same result as the "ordinary" derivative for polynomials. Whether "exactly identical to" implies "a surprising amount to do with" depends on how easily surprised you are, I guess. ... I glossed over the sense in which the "primes" are precisely the linear polynomials, so here are a few words about that for anyone who's curious. If we look at polynomials with complex-number coefficients, a beautiful theorem says that they can all be written as A (x-r1) (x-r2) ... (x-rk), and then one polynomial divides another if and only if its set of rj is a subset of the other's (handling repeated roots in the "obvious" way). It's pretty easy to get from this that the linear polynomials are (1) the irreducible ones, i.e., the ones that can't be factored into lower-degree polynomials, and (2) the prime ones, i.e., the ones with the property that if p divides ab then p divides either a or b. (These properties are equivalent for the integers, as well as for polynomials with complex coefficients, but there are other settings in which they come out different, and both of them are useful, so they have different names.) (What happens if we use real rather than complex coefficients? The Wikipedia "Arithmetic derivative" page claims that we still get the usual derivative, but that looks wrong to me, because if we work over the real numbers then x^2+1 is both prime and irreducible, but its derivative isn't 1. Maybe I'm missing something.)