The $\mu_{1},\mu_{2}$ derivatives are clear. For the $\bar{\eta}$ derivative, we can apply the component form of the distributional equation (2.4) away from $\mathfrak{D}$, to see

$$\begin{split}\frac{\partial\beta_{1}}{\partial\bar{\eta}}&=2\lim\int\frac{\partial^{2}v^{11}}{\partial\eta\partial\bar{\eta}}d\mu_{1}+\frac{\partial^{2}v^{12}}{\partial\eta\partial\bar{\eta}}d\mu_{2}\\ &=-\frac{1}{2}\lim\int\frac{\partial^{2}w}{\partial\mu_{1}\partial\mu_{1}}d\mu_{1}+\frac{\partial^{2}w}{\partial\mu_{1}\partial\mu_{2}}d\mu_{2}=-\frac{1}{2}\frac{\partial w}{\partial\mu_{1}},\end{split}$$

where in the last equality we compare the asymptotic values at infinity to show there is no constant term depending on $\eta$. Likewise with $\beta_{2},\beta_{0}$. ∎