  An arithmetic function $f$ is Leibniz-additive if there is a completely
multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for
all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$
for all positive integers $m$ and $n$. A motivation for the present study is
the fact that Leibniz-additive functions are generalizations of the arithmetic
derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper,
we study the basic properties of Leibniz-additive functions and, among other
things, show that a Leibniz-additive function $f$ is totally determined by the
values of $f$ and $h_f$ at primes. We also consider properties of
Leibniz-additive functions with respect to the usual product, composition and
Dirichlet convolution of arithmetic functions.
