I need to prove that Constant Absolute Risk Aversion (CARA) is equivalent to \begin{gather} \int u'(x)dF(x) = u'(c(F,u)) \end{gather}
where $u(x)$ is a Bernoulli utility function, $F$ is the distribution of the lottery and $c(F,u)$ is the certainty equivalent.
I started from the fact that CARA is defined as $-\frac{u''(x)}{u'(x)}=a$, where $a$ is a constant and that the certainty equivalent is defined as $u(c(F,u))=\int u(x)dF(x)$.
I tried to mix up the two definitions but I am a bit lost with the interaction of integration and differentiation.
Do you have any hint?