A ring K is BMCR (Brown-McCoy ring) if the prime radical is the same as the Brown-McCoy radical in every homomorphic image of K. It is known (Watters) that K(x) is BMCR\(\leftrightarrow K\) is BMCR. If D is a derivation of K then we say K is D-BMCR if the D-prime radical is the same as the D-Brown-McCoy radical in every homomorphic image of K. If we call \(K[x,D]=R\) then in general it is not true that K is D- BMCR\(\leftrightarrow R\) is BMCR. The author proves that K is D-BMCR if and only if 1) R is BMCR and 2) For every maximal ideal M of R, K/(M\(\cap K)\) is a D-simple ring with 1. Some connections are made between the prime and D-prime ideals in differential intermediate extensions of liberal extensions. The paper closes with several examples.