= Solution
Build a narrow-band <wave packet> from <plane waves> with <wavenumbers> close to $k_0$:
$$
\psi(x,t)=\int A(k)e^{i[kx-\omega(k)t]}\,dk.
$$
The <Taylor expansion> of the <dispersion relation> is $\omega(k)\simeq\omega_0+\omega'_0(k-k_0)$, where $\omega'_0=(d\omega/dk)_{k_0}$. Put $\kappa=k-k_0$ to obtain
$$
\psi(x,t)\simeq e^{i(k_0x-\omega_0t)}
\int A(k_0+\kappa)e^{i\kappa(x-\omega'_0t)}\,d\kappa
=e^{i(k_0x-\omega_0t)}f(x-\omega'_0t).
$$
The carrier's <wave phase> travels at the <phase velocity> $\omega_0/k_0$, while the envelope has constant shape at constant $x-\omega'_0t$. Its <group velocity> is therefore
$$
\boxed{v_g=\left.\frac{d\omega}{dk}\right|_{k_0}.}
$$
The approximation neglects higher derivatives of the <dispersion relation>. Those derivatives cause the <wave packet> to spread or distort over sufficiently long propagation times; the result describes the local envelope speed of a sufficiently narrow band.
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