We don't need addition for the Leibniz rule

We show that for a linear differentiation, the Leibniz rule is equivalent to the condition that all its exponentials lie on a single ray in a vector space. We generalize the notions of a multiplicative exponential and a derivative to right rings and establish conditions under which the kernel of a differentiation embeds into the constants. We prove an analogue of the homomorphism theorem for differentiations and give an explicit form of certain differentiations of a triangular rring that lie beyond the scope of the known classification.