The softening of the radial breathing mode (RBM) of metallic single walled carbon nanotubes (m-SWNTs) due to electron-phonon coupling has been studied by observing the Fermi level [see text] dependence of the RBM Raman peak. In situ Raman spectra were obtained from several individual m-SWNTs while varying [see text] electrochemically. The RBM frequency of an intrinsic m-SWNT is shown to be down-shifted relative to highly doped tubes by approximately 2 cm(1). The down-shift is greatest for small diameter and small chiral angle SWNTs. Most tubes show no change in RBM linewidth. A comparison is drawn between the RBM and the G band (A(LO) phonon) with respect to the [see text] dependence of their frequencies and linewidths.