= Solution
At fixed <particle number> and <pressure>, the joint <isothermal-isobaric ensemble> weight for a <microstate> of <energy> $E$ and <volume> $V$ is $e^{-\beta(E+PV)}$. Write its <enthalpy> as $H=E+PV$. Differentiating the <isothermal-isobaric partition function> with a temperature-independent <volume> measure gives
$$
\partial_\beta Q_P=-\int_0^\infty dV\sum_s H_s(V)e^{-\beta H_s(V)}.
$$
After division by $Q_P$ this is the normalized <ensemble average>, so
$$
\boxed{\langle H\rangle_P=-\partial_\beta\ln Q_P}.
$$
The differentiation assumes that the microscopic <Hamiltonian> and the integration measure have no additional explicit <temperature> dependence. An extra temperature-dependent normalization factor must be differentiated separately.
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