 Group velocity from a period-dependent phase velocity

Since $k=\omega /c(\omega )$, differentiation gives $u=(dk/d\omega {)}^{-1}=c/[1-(\omega /c)dc/d\omega ]$. Using the period $P=2\pi /\omega$ gives the displayed formula. A positive period slope makes the group velocity smaller than the phase velocity on this branch. A slope read from a graph produces an approximate local result, not a globally fitted dispersion law.