Computation of option vega under CEV

It is easy to define the option vega $\nu=\frac{\partial C}{\partial \sigma}$ under Black Scholes model since volatility is a single quantity.

However, under CEV or local volaility model, it is confusing for me to compute option vega.

For example, the volaility function is defined as $\sigma(S)=\delta S^{\beta}$. Then, how to compute a sensitivity of the option price with respect to $\sigma(S)$??

At first galance, I think $$\nu = \frac{\partial C}{\partial \sigma(S)}= \frac{\partial C}{\partial S}\frac{\partial S}{\partial \sigma(S)}=\Delta\frac{1}{\delta\beta S^{\beta-1}}$$

Is this right? But it seems wrong..