Derivation with divided Leibniz condition powers

Usual notion: derivation with divided Leibniz condition powers for all nonnegative integers
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring). A derivation with divided Leibniz condition powers for $$R$$ is the following data:

satisfying the following Leibniz-type compatibility condition (divided version of binomial formula for powers of a derivation):

For all nonnegative integers $$n$$, we have $$d^{(n)}(x * y) = \sum_{i + j = n} d^{(i)}(x) * d^{(j)}(y)$$

Note that $$d^{(1)} = d$$ must be a derivation in the usual sense of the word.

Derivation with divided Leibniz condition powers up to a point
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring) and $$m$$ is a natural number. A derivation with divided Leibniz condition powers up to $$m$$ for $$R$$ is the following data:

satisfying the following Leibniz-type compatibility condition (divided version of binomial formula for powers of a derivation):

For all nonnegative integers $$n \le m$$, we have $$d^{(n)}(x * y) = \sum_{i + j = n} d^{(i)}(x) * d^{(j)}(y)$$

Note that $$d^{(1)} = d$$ must be a derivation in the usual sense of the word.

Facts

 * Exponential of nilpotent derivation with divided Leibniz condition powers is endomorphism

Related notions

 * Derivation with divided powers is a stronger condition that insists that the $$d^{(n)}$$ also form a system of divided powers of the derivation.