Algebras of constants for some extensions of derivations

Let A = B[t] be the polynomial ring in one variable over a unique factorization domain B, where k is a field of characteristic zero. Let delta be a derivation of B, phi is an element of B with delta(phi) not equal 0, and let s be a positive integer. Denote by d the derivation of A such that d(b) = t(s)delta(b) for b is an element of B and d(t) = 1/s+1 delta(phi). We describe the rings of constants of d. Several applications of our description are also given.