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Write the exponential dispersion family in the conventionAssume that the support does not depend on the parameters, that $\theta$ is in the interior of the natural parameter space, and that differentiation can pass under the normalizing integral or sum. Differentiating $\int f=1$ at fixed dispersion parameter givesso the expectation is $b^{\prime }(\theta )$. A second differentiation givesConsequentlyHere $b$ is the cumulant function of an exponential family; the same identities follow by differentiating $\log \mathbb{E}{e}^{tZ}=[b(\theta +\varphi t)-b(\theta )]/\varphi$. For a known observation weight $w$, replace $\varphi$ by $\varphi /w$. These regularity assumptions for differentiation under the integral sign matter: differentiation of a parameter-dependent support can introduce additional boundary terms.