partial derivative of the Hamiltonian inside a function Let $f( \mathcal{H}(q,p))$ where $q(t)$ and $p(t)$ are time dependent.
My Professor wrote:
$$
\frac{\partial f}{\partial t} = f' \frac{\partial H}{\partial t}
$$$$
\frac{\partial f}{\partial q} = f' \frac{\partial H}{\partial q}
$$
$$
\frac{\partial f}{\partial p} = f' \frac{\partial H}{\partial p}
$$
I'm not sure if my professor doesn't take into consideration the derivatives of $f$ and just calls them $f'$. 
In other words is he saying that $f' = d_tf = d_qf = d_pf$. or am I missing something ?
 A: It's the usual confusion. Slightly oversimplified explanation is the following: he is using $f$ for two different functions. One is $f:\mathbb{R}\to \mathbb{R}$. The other is the composition of this $f$ with $H$. We'll denote it by $F=f\circ H$. Then $F$ is a function of $p, q, t$ just like $H$, while $f$ is a function of single input, which is not named but we can call $u$. Then $f'$ is the derivative of $f$ with respect to the only input it has, i.e. $u$, while "$\frac{\partial f}{\partial q}$" should really be "$\frac{\partial F}{\partial q}$" and is by chain rule
$$\frac{\partial F}{\partial q}=\frac{\partial (f\circ H)}{\partial q}= \frac{df}{du}\frac{\partial H}{\partial q}=f'\frac{\partial H}{\partial q}$$
In fact, in the setup you describe the full story is more complicated. There are 2 functions both denoted by $H$ - one is a function of two inputs (we'll call it $H$) and one is a function of time only, namely what we will call $h(t)=H(p(t), q(t))$. These are different and are commonly denoted by the same letter (in your prof's case by $H$).
There are 3 functions all called $f$: the function of single input, the function of two inputs $F=f\circ H$ and the function of time only, $\phi$:
$\phi(t)=f(h(t))=F(p(t), q(t))=f(H(p(t), q(t)))$.
These are all different and are commonly denoted by the same letter (in your prof's case by $f$).
All partial derivatives of $f, F, \phi$ and $h, H$ are interconnected by chain rule in various ways, similarly to the above, but slightly more complicated.
The "usual confusion" arises when one uses the same letter for all the functions and thus for their partial derivatives. This is very common everywhere, but especially in physics, where the variables have some meaning (like $H$ is total energy), and it seems weird to give them different name when considering them as functions of one set of inputs vs another: say, $h$ when considered as function of time only, versus $H$ when considered as function of position and momentum. I mean, energy is energy, it's all $H$ to a physicist. Nevertheless, mathematically these are different functions with different domains (and as a result different types of partials), and when doing math with them this sometimes comes up.
(The "this notation sucks" rant continued: Even worse, the notation for partial derivatives interplays with this in exactly the wrong way -- by providing the info about which input is being varied, and mostly ignoring the question of what is being held fixed. This becomes particularly unfun when you do thermodynamics, but that's another story.)
