Given a dynamical system with canonical variables $q$ and $p$. The equations of motion are given by Hamilton's equations
\begin{equation} \dot{q}=\frac{\partial H}{\partial p} \\ \dot{p}=-\frac{\partial H}{\partial q} \end{equation}
where a dot represents a derivative respect to time and $H=H(q,p)$ is the hamiltonian of the system. I'm going to use this as an example to show what my question is, but my concern is more general. I understand that the derivatives on the rhs of the equations are not taken along the trajectories. If they were, any system with constant energy would give identically zero, since $H=$ constant. However, the lhs are evaluated on the trajectories since the canonical variables $q$ and $p$ have time dependence. This means that the right hand side is supposed to be evaluated on the phase space trajectory $\big(q(t),p(t)\big)$after we apply the derivative. Why is this not explained in any book (Goldstein, Lanczos, etc.) that I have encountered when studying Classical Mechanics? Is it that obvious?
This same problems appears when dealing with the Poisson brackets. When we calculate
\begin{equation} \{q,p\}=\frac{\partial q}{\partial q}\frac{\partial p}{\partial p}-\frac{\partial q}{\partial p}\frac{\partial p}{\partial q}=1-0=1 \end{equation}
I assume that we are using $q$ and $p$ as canonical variables, without any time dependence. In other words, the variables are not evaluated along the solutions of Hamilton's equations. However, when we write Hamilton's equations using Poisson brackets
\begin{equation} \dot{q}=\{q,H\}=\frac{\partial q}{\partial q}\frac{\partial H}{\partial p }- \frac{\partial q}{\partial p }\frac{\partial H}{\partial q}= \frac{\partial H}{\partial p} \end{equation}
The Poisson bracket is clearly evaluated at the trajectories after we apply the derivatives, just like Hamilton's equations.
However, I have also encountered terms like $\frac{\partial q}{\partial p}$ that were actually evaluated at the trajectory before taking the derivative. Specially when dealing with changes of coordinates and Canonical transformations. So the question is: Is there a way of indicating when derivatives are applied before evaluating on the trajectory and when they are applied after that?
