  The original 2017 version of this paper, published in Ann. Appl. Probab., 27,
1678--1801, contains a major gap in the proofs. In the subsequent publication
in Ann. Appl. Probab., 34, 3370--3374, 2024, we indicated how to fix this. For
convenience of the reader, we here update the original paper to incorporate the
suggested fix.
  Consider a Boolean model in $R^d$ with balls of random, bounded radii with
distribution $F_0$, centered at the points of a Poisson process of intensity
$t>0$. The capacity functional of the infinite cluster $Z_\infty$ is given by
$\theta_L(t) = P(Z_\infty\cap L \neq \emptyset)$, defined for each compact
$L\subset R^d$. We prove for any fixed $L$ and $F_0$ that $\theta_L(t)$ is
infinitely differentiable in $t$, except at the critical value $t_c$; we give a
Margulis-Russo type formula for the derivatives.
  More generally, allowing the distribution $F_0$ to vary and viewing
$\theta_L$ as a function of the measure $F:=tF_0$, we show that it is
infinitely differentiable in all directions with respect to the measure $F$ in
the supercritical region of the cone of positive measures on a bounded
interval.
  We also prove that $\theta_L(\cdot)$ grows at least linearly at the critical
value. This implies that the critical exponent known as $\beta$ is at most 1
(if it exists) for this model. Along the way, we extend a result of H.Tanemura
(1993), on regularity of the supercritical Boolean model in $d \geq 3$ with
fixed-radius balls, to the case with bounded random radii.
