Prove (Something) is Differentiable

I recently took an exam where it was implied (but not stated or asked to be shown) that a function $\displaystyle \psi(\theta)$ satisfying

$\displaystyle \displaystyle

\int_{\mathbb{R}} \exp[\theta x - \psi(\theta) + h(x)] \ dx = 1$

is differentiable. I think you can show this using a dominated convergence theorem, but I haven't taken a course in analysis that provided the machinery to do this rigorously. Some proof (or counterexample) would be appreciated. If it requires a theorem I haven't seen, that's fine.

If anyone's curious, the application is in statistics - ultimately the goal is to show that the maximum likelihood estimator of $\displaystyle \theta$ is asymptotically normal.