The log-likelihood of any exponential family is simply $n[\bar{\s} \tp \bt - \psi(\bt)]$ plus a constant, so
\[\begin{align*} \u &\propto \bar{\s} - \nabla \psi(\bt) \\ \bth &= (\nabla\psi)^{-1}(\bar{\s}) \end{align*}\]Since the MLE is simply a function of the mean in exponential families, it is particularly easy to derive its limiting distribution. Letting $\bm = \Ex(\s)$, the central limit theorem tells us that
\[\sqrt{n}(\bar{\s} - \bm) \inD \Normal(\zero, \V),\]where $\V = \nabla^2 \psi(\bt)$. Thus, letting $\g$ denote the transformation $\bt=\g(\bm)$, we have, by the delta method:
\[\sqrt{n}(\bth - \bt) \inD \Normal(\zero, \nabla \g (\bm) \tp \V \nabla \g (\bm));\]keep in mind here that $\nabla \g$ and $\V$ are both $d \times d$ matrices.
See also: Likelihood: Asymptotic normality.