 Secant domination for expected utility derivatives

If a negative differentiable concave utility function satisfies $\mathbb{E}U(\theta X)>-\infty$ on an open interval, outer secants at four points bound the sample derivative on a smaller compact interval. Negativity makes the endpoint values absolutely integrable. The dominated convergence theorem then gives $d\mathbb{E}U(\theta X)/d\theta =\mathbb{E}[X{U}^{\prime }(\theta X)]$ and continuous derivative when $U^{\prime }$ is continuous. This is a justification of differentiation, rather than an assumption of integrability of marginal utility.