In many books on nonlinear fiber optics, the Taylor series expansion of the mode-propagation constant $\beta$ is performed about a frequency $\omega_0$ at which a pulse's spectrum is to be centered.
$\beta(\omega) = n(\omega)\frac{\omega}{c} = \beta_0 + \beta_1(\omega - \omega_0) + \frac{1}{2}\beta_2(\omega - \omega_0)^2 + ...$
where
$\beta_m = (\frac{d^m\beta}{d\omega^m})_{\omega = \omega_0} (m = 0,1,2,...)$
Now the authors say that the parameters $\beta_1$ and $\beta_2$ are related to the refractive index and its derivatives through
$\beta_1 = \frac{1}{v_g} = \frac{n_g}{c} = \frac{1}{c}(n + \omega\frac{dn}{d\omega})\\$
$\beta_2 = \frac{1}{c}(2\frac{dn}{d\omega} + \omega\frac{d^2n}{d\omega^2})$
Can someone explain how these relations can be found?