Taking volatility swap payoff as $$( \sigma_F - \sigma_S ) * volatility~notional $$
and Taking variance swap payoff as $$( \sigma_F^2 - \sigma_S^2 ) * variance~notional $$
I am trying to understand the origin of the relationship;
$$variance~notional = \frac{vega}{(2\sigma_s)}$$
I understand that vega is volatility notional as $\frac{\delta f} {\delta \sigma_F}$ is the change of payoff with respect to volatility point
I understand that variance notional is $\frac{\delta f} {\delta \sigma_F^2}$ as this is the change of payoff with respect to variance point
and $2\sigma_s$ is obviously the derivative of $\sigma_s^2$