= Solution
In <canonical coordinates> on <phase space>, the <Poisson bracket> is
$$
\boxed{\{f,g\}=\sum_{i=1}^n\left(
\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}
-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right).}
$$
It treats $t$ as a parameter. Along a solution of <Hamilton's equations>, the <chain rule> gives
$$
\frac{df}{dt}=\sum_i\left(f_{q_i}\dot q_i+f_{p_i}\dot p_i\right)+f_t
=\sum_i\left(f_{q_i}H_{p_i}-f_{p_i}H_{q_i}\right)+f_t.
$$
Therefore the <Hamiltonian conservation law> is
$$
\boxed{\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}.}
$$
The total <derivative> follows the trajectory; the explicit <partial derivative> holds $(\mathbf q,\mathbf p)$ fixed.
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