An arithmetic function 𝑓 is Leibniz-additive if there is a completely multiplicative function ℎ𝑓 such that 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ𝑓 (𝑛) + 𝑓(𝑛)ℎ𝑓 (𝑚) for all positive integers 𝑚 and 𝑛. A motivation for the present study is the fact that Leibnizadditive functions are generalizations of the arithmetic derivative 𝐷; namely, 𝐷 is Leibnizadditive with ℎ𝐷(𝑛) = 𝑛. 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 ℎ𝑓 at primes. We also find connections of Leibniz-additive functions to the usual product, composition and Dirichlet convolution of arithmetic functions. The arithmetic partial derivative is also considered.