We measure resonant Raman scattering from 11 individual C60-derived double-wall carbon nanotubes all having inner semiconducting (6,5) tubes and various outer metallic tubes. The Raman spectra show the radial breathing modes (RBM) of the inner and the outer tubes to be simultaneously in resonance with the same laser energy. We observe that an increase in the RBM frequency of the inner tubes is related to an increase in the RBM frequency of the outer tubes. The Raman spectra also contain a sharp G− feature that increases in frequency as the nominal diameter of the outer metallic tubes decreases. Finally, the one-phonon second-order D-band mode shows a two-way frequency splitting that decreases with decreasing nominal wall-to-wall distance. We suggest that the stress which increases with decreasing nominal wall-to-wall distance is responsible for the hardening that is observed in the frequencies of the RBM, D and G− modes of the inner (6,5) semiconducting tubes.