Choosing for every prime number $p$ an arbitrary rational value $\alpha_p$ for its formal derivation $p'$, one gets a derivation on the rationals by extending the definition with the Leibniz rule given by $\left(\pm\prod_{p}p^{\nu_p}\right)'=\pm\sum_{p}\alpha_p\frac{\nu_p}{p} \prod_{p}p^{\nu_p}$ where sums and products are over all primes (and one can even consider rational exponents $\nu_p$ but let us stick to integers for simplicity). We have then $(ab)'=a'b+ab'$ and $(a^n)'=na^{n-1}a'$ but $(a+b)'\not=a'+b'$ in general.

Are there choices of the values $\alpha_p$ giving rise to interesting dynamical systems $x\longrightarrow x'$ on the rationals? (The word "interesting" is of course not well-defined but examples are: only finitely many orbits (in the sense that $a,b$ are considered in the same orbit if $a^{(n)}=b^{(m)}$ for natural integers $n,m$), almost all orbits $a^{(\mathbb N)}$ have infinitely many different elements, almost all orbits $a^{(\mathbb N)}$ are finite, ...)

Natural candidates are perhaps $\alpha_p=1$ for all $p$, $\alpha_p$ a (linear combination of) Dirichlet character(s) with rational values or for example $\alpha_p=p$ (this choice fixes all primes). I guess that in general one cannot say much and the study is messy, except if all $\alpha_p$ except one are zero. A first perhaps non-trivial case is given by $\alpha_p=0$ for all primes $p\geq 5$ but $\alpha_2$ and $\alpha_3$ are both non-zero rationals.

Which choices of values for $\alpha_2,\dots$ give rise to an interesting differential calculus?

More precisely, we want a canonical primitive in $\mathbb Q[l_2,l_3,l_5,l_7,l_{11},\dots]$ for every rational number where $l_p$ is a symbol for the logarithm $\int p^{-1}$ of each prime $p$. Are there such choices and if yes has the map $x\longrightarrow \int x$ an interesting dynamical behaviour?

I guess that this can be extended to general number fields (where one has probably to work with ideals instead of numbers).

Motivation: none except fun and perhaps the existence of a good exercise on the Leibniz rule and the chain rule for undergraduates.

Journal of Integer Sequenceson this topic. Keywords: "number derivative". I don't recall your questions specifically being addressed, but nevertheless you might want to take a look. – Pete L. Clark May 26 '10 at 8:28