  We first introduce the arithmetic subderivative of a positive integer with
respect to a non-empty set of primes. This notion generalizes the concepts of
the arithmetic derivative and arithmetic partial derivative. More generally, we
then define that an arithmetic function $f$ is Leibniz-additive if there is a
nonzero-valued and completely multiplicative function $h_f$ satisfying
$f(mn)=f(m)h_f(n)+f(n)h_f(m)$ for all positive integers $m$ and $n$. We study
some basic properties of such functions. For example, we present conditions
when an arithmetic function is Leibniz-additive and, generalizing well-known
bounds for the arithmetic derivative, establish bounds for a Leibniz-additive
function.
