Functionalized dendrimers are a novel type of dendritic macromolecules that contain functionalized groups that do not .take part in the dendrimer's synthetic cycle. Focusing on the local dynamics of the functionalized dendrimers, we study NMR relaxation by considering the spin-lattice relaxation rate 1/T-1 (omega) of the H-1 and C-13 nuclei and the corresponding spectral density. The macromolecules are modeled in the framework that accounts for the local bending rigidity, which is of paramount importance for the NMR relaxation of dendritic structures. We show that the behavior of the NMR relaxation of the functionalized dendrimers is qualitatively different from that of the standard dendrimers manifested through an additional maximum in omega/T-1(omega). The new theoretical picture has allowed us to analyze recent atomistic molecular dynamics simulations of carbosilane dendrimer melts and to elucidate the differences in the high-frequency behavior of their spin-lattice relaxation rates from the predictions of the standard dendrimer model.