Solution
= Solution
For a phase-space function with no explicit time dependence, the <chain rule> and <Hamilton's equations> give
$$
\frac{df}{dt}=\sum_i(\partial_{q_i}f\,\dot q_i+\partial_{p_i}f\,\dot p_i)=\sum_i(\partial_{q_i}f\,\partial_{p_i}H-\partial_{p_i}f\,\partial_{q_i}H)=\boxed{\{f,H\}}.
$$
An explicitly time-dependent function would instead satisfy $df/dt=\partial_tf+\{f,H\}$.