= 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 $v_g=v_p-\lambda_m\,dv_p/d\lambda_m$. This follows by differentiating $\omega=kv_p$ and using $k\,d\lambda_m/dk=-\lambda_m$. If instead the independent variable is vacuum wavelength $\lambda_0=2\pi c/\omega$, then $k=\omega/v_p(\lambda_0)$ gives
$$
\boxed{v_g=\frac{v_p}{1+(\lambda_0/v_p)\,dv_p/d\lambda_0}
=\frac{c}{n-\lambda_0\,dn/d\lambda_0}}.
$$
Substituting 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.
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