We demonstrate that the electronic band structure extracted from quasiparticle interference spectroscopy [Nat. Phys. 9, 468 (2013)1745-247310.1038/nphys2671] and the theoretically computed form of the superconducting gaps [Proc. Natl. Acad. Sci. USA 111, 11663 (2014)PNASA60027-842410.1073/pnas.1409444111] can be used to understand the dI/dV line shape measured in the normal and superconducting state of CeCoIn5 [Nat. Phys. 9, 474 (2013)1745-247310.1038/nphys2672]. In particular, the dI/dV line shape, and the spatial structure of defect-induced impurity states, reflects the existence of multiple superconducting gaps of dx2-y2 symmetry. These results strongly support a recently proposed microscopic origin of the unconventional superconducting state. © 2016 American Physical Society.