How to derive group velocity dispersion (GVD)?

I’m trying to arrive to the following expression to group velocity dispersion:

$$text{GVD} = frac{lambda^3}{2 pi c^2} frac{d^2n}{d omega^2}.$$

I know that GVD can be expressed as:

$$text{GVD} = frac{d}{domega}left( frac{1}{v_g} right)$$

where

$$frac{1}{v_g} = frac{dk}{domega}
k = frac{omega}{c} n(omega)
omega = frac{2pi c}{lambda}$$

and the refraction index depends on the wavelength.

How can I arrive at the first GVD expression knowing the others?