  We investigate the properties of arithmetic differentiation, an attempt to
adapt the notion of differentiation to the integers by preserving the Leibniz
rule, (ab)' = a'b + ab'. This has proved to be a very rich topic with many
different aspects and implications to other fields of mathematics and
specifically to various unproven conjectures in additive prime number theory.
Our paper consists of a self-contaited introduction to the topic, along with a
couple of new theorems, several of them related to arithmetic differentiation
of rational numbers, a topic almost unexplored until now.
