Motivation
Diophantine approximation and Nevanlinna theory share remarkable similarities. What is missing on the number theory side is a proper derivative. There is the arithmetic derivative, an operation on numbers, which satisfies the product rule but not linearity. The proof of the second main theorem in Nevanlinna theory uses the fact that for a function $f$ and a number $a$, $f' = (f-a)'$. This seems impossible to achieve for the arithmetic derivative. I would like to know if a weaker result holds (or why it fails)?
Definitions
For $x=\prod_{i} p_i^{e_i} \in \mathbb{Q}$ define the arithmetic derivative $x'=x\times \sum_{i} e_i \frac{p_i'}{p_i}$ with $p_i' \in \mathbb{Z}$. The usual arithmetic derivative has $p_i'=1$, but we don't yet fix the $p_i'$. Also define $0'=0$. Then $(xy)' = x'y+y'x$ and it follows that $1'=0$.
For $x \in \mathbb{Q}$, written as $x=p^e\frac{r}{s}$ with $r,s \in \mathbb{Z}$ and where $p$ does not divide $rs$, define the p-adic absolute value $||x||_p = p^{-e}$.
Question
Let $x, a_1,\ldots,a_k \in \mathbb{Q}$. Is it possible to define an arithmetic derivative (which is allowed to depend on $x$ and the $a_j$) by fixing the $p_i'$ such that $||x'||_p \leq ||(x-a_j)'||_p$ holds for all $p$ and all $a_j$ with finitely many exceptions (which may depend on the $a_j$ but not on $x$)?