  We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the
Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a
suitable non-degeneracy condition. Using Geometry of Numbers, we
unconditionally show their existence with controlled size. We prove that any
power-saving improvement on our size bounds would give a version of the $abc$
Conjecture. In fact, we show that the existence of sufficiently small
arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our
results give an explicit manifestation of an analogy suggested by Vojta in the
eighties, relating Geometry of Numbers in arithmetic to derivatives in function
fields and Nevanlinna theory. In addition, our construction formalizes the
widespread intuition that the $abc$ Conjecture should be related to arithmetic
derivatives of some sort.
