= Solution
Let $s=Kx$ and assume $s>0$. The scalar <Poisson data fidelity> integrand has derivative $1-y/s$. Applying the permitted interchange of differentiation and integration in a direction $z$ gives
$$
\left.\frac d{d\varepsilon}E(x+\varepsilon z)\right|_{\varepsilon=0}=\int_\Sigma\left(1-\frac y{Kx}\right)Kz\,dt=\left\langle z,K^*\left(1-\frac y{Kx}\right)\right\rangle.
$$
Thus the ambient-space <Fréchet derivative>, or its <Hilbert space> gradient when the pairing is an inner product, is
$$
\boxed{E'(x)=K^*\left(1-\frac y{Kx}\right).}
$$
The quotient is pointwise, and the <adjoint operator> transports it back from the data domain to the source domain. The chosen spaces must ensure the integrals and derivative pairing are finite; an output bounded away from zero is a useful sufficient condition.
The printed set of <probability density functions> and positive cone are not vector spaces, so the notation for a bounded linear map between them is understood as a positive <bounded linear operator> on ambient function spaces, restricted to admissible inputs. If unit mass is a constraint, admissible directions satisfy $\int_\Omega z=0$. The displayed gradient is then a representative modulo an additive constant; optimization with that constraint additionally includes its <normal cone>.
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