Undergrad Derivation of phase change parameter in dispersive medium

[VittorioT]

Hi, I'm trying to evaluate the derivates of first, second and third order of the phase change parameter in a dispersive medium.
In such medium the refractive index is a function of the wavelength.
In my case it depends on the wavelength in vacuum.

\begin{equation*} n(\lambda_0 )\end{equation*} and it has a known expression that I can easy derivate in terms of the wavelength in vacuum.

\begin{equation*}
\beta =\frac{\omega } cn(\lambda_0 )
\end{equation*}
\begin{equation*}
\frac{\partial \beta }{\partial \omega }=\frac{\partial }{\partial \omega }[\frac{\omega } cn(\lambda_0 )]=\frac 1 cn(\lambda_0 )+\frac{\omega } c\frac{\partial }{\partial \omega }[n(\lambda_0 )]
\end{equation*}

Before I could write this:
\begin{equation*}
\lambda_0 =\frac{2\pi c}{\omega }
\end{equation*}
but in general:
\begin{equation*}
\lambda =\frac{2\pi c}{\omega n}
\end{equation*}
or even maybe in this case:
\begin{equation*}
\lambda =\frac{2\pi c}{\omega n(\lambda_0 )}
\end{equation*}
Which one of the last three equation do I have to differentiate in order to proceed with derivatives?

 

[blue_leaf77]

Remember, $\lambda_0$ is the wavelength in vacuum.