Why is a vertex a derivative of the propagator? Where can I find the proof to this nice trick:
if the momentum $q$ is small, the vertex is the derivative with respect to the mass of a propagator times a factor $(-m/v)$ like in the picture:

 A: First note that
\begin{equation}
\frac{\partial}{\partial m} \frac{i}{\not p - m} = \frac{i}{(\not p-m)^2}
\end{equation}
Now the exact answer for finite $q$ for that diagram is
\begin{equation}
\mathcal{A} = \left( \frac{im}{v} \right) \frac{i}{\not p-m}  \frac{i}{\not p + \not q - m} G(q)
\end{equation}
where $G(q)$ is the propagator for whatever particle is on the dotted line (I'm guessing it's a higgs).
Now for whatever reason they've stripped the dotted propagator for the answer$^1$, let's just take that as give, so we are really interested in
\begin{equation}
\mathcal{A}' = \frac{im}{v} \frac{i}{\not p- m} \frac{i}{\not p + \not q -m}
\end{equation}
 In the limit $q\rightarrow 0$, the above becomes
\begin{equation}
\mathcal{A}' = \frac{im}{v} \left(\frac{-1}{(\not p-m)^2} + O (q)\right) \rightarrow \frac{-m}{v} \frac{i}{(\not p-m)^2} = \frac{-m}{v} \frac{\partial}{\partial m} \frac{i}{\not p - m}
\end{equation}
which is what you wanted.
The overall point is that things simplify in the $q\rightarrow 0$ limit.

$^1$ I'm not 100% sure why they do this without thinking about it more, but on a quick glance through the notes it looks like they have in mind processes like $H\rightarrow \gamma \gamma$, so in the identity you use the Higgs is implicitly taken to be an external particle, whereas the fermions run in a loop. Thus the Higgs propagator will disappear since it is an external line in the diagram you are ultimately interested in, whereas the fermion propagators show up in the loop. The net result of this trick is then to effectively convert a three point function at one loop to the derivative of a two point function at one loop, which is much easier to handle. Again, physically, the idea is that as the momentum becomes very small the calculation should simplify. This kind of trick is very useful.
