= Solution
Since $H$ is a <conserved quantity>, the <Hamiltonian energy balance> gives $H_t=0$. Since $f$ is a <conserved quantity>, $f_t+\{f,H\}=0$. Take an explicit <partial derivative> with respect to time, holding the <phase space> coordinates fixed:
$$
f_{tt}+\{f_t,H\}+\{f,H_t\}=0.
$$
The last <Poisson bracket> vanishes because $H_t=0$. Thus
$$
\boxed{\frac{d}{dt}\left(\frac{\partial f}{\partial t}\right)
=f_{tt}+\{f_t,H\}=0.}
$$
This is the <time derivative of a conserved quantity for an autonomous Hamiltonian>. The conservation hypotheses are understood to hold for every trajectory, as identities on <phase space>.
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