 Group velocity in material and vacuum wavelength coordinates

Writing the material wavelength as ${\lambda }_m=2\pi /k$ and its phase velocity as $v_p({\lambda }_m)$ gives $v_g=v_p-{\lambda }_{m}d{v}_p/d{\lambda }_m$. This follows by differentiating $\omega =k{v}_p$ and using $kd{\lambda }_m/dk=-{\lambda }_m$. If instead the independent variable is vacuum wavelength ${\lambda }_0=2\pi c/\omega$, then $k=\omega /v_p({\lambda }_0)$ givesSubstituting vacuum wavelengths into the material-wavelength derivative formula gives an erroneous result beyond first order in dispersion. The two coordinates describe the same dispersion relation, but their derivatives differ by the chain rule.