Derivations of skew Ore polynomial semirings

We investigate derivations in the semiring of skew Ore polynomials over an idempotent semiring. We show that multiplying each polynomial by x on left is a derivation and construct commutative idempotent semiring consisting of derivations of a skew polynomial semiring. We introduce generalized hereditary derivations as derivations acting only over the coefficients of the polynomial and construct an S-derivation in the classical sense of Jacobson. Finally, we give a description of the δ-derivations in a skew polynomial semiring S[x] and show that an arbitrary δ-derivation can be represented by a generalized hereditary derivation and an S-derivation.