= Solution
By the multivariable <chain rule> and <Hamilton's equations>,
$$
\frac{df}{dt}
=\sum_i\left(
\frac{\partial f}{\partial q_i}\dot q_i
+\frac{\partial f}{\partial p_i}\dot p_i
\right)+\frac{\partial f}{\partial t}
=\sum_i\left(
\frac{\partial f}{\partial q_i}\frac{\partial H}{\partial p_i}
-\frac{\partial f}{\partial p_i}\frac{\partial H}{\partial q_i}
\right)+\frac{\partial f}{\partial t}.
$$
The sum is the <Poisson bracket> $\{f,H\}$, giving
$$
\boxed{\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}.}
$$
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