Arithmetic subderivatives and Leibniz-additive functions

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.