derivative at a given point I often see the following:
$$ \left. \frac{\partial q}{\partial \alpha} \right|_{\alpha = 0} $$
Where $q$ is a function $q(q', t, \alpha)$.
Is that just the same as that?
$$ \frac{\partial}{\partial \alpha} q(\alpha=0) $$
If they are the same, why do people write the first one? I find the second easier.
 A: I don't know if I can add much that is helpful to what Dirk has written, but: You understanding is correct. The two expressions mean the same thing. It is indeed just a matter of taste.
The vertical line $\bigg\lvert_{\alpha = 0}$ simply means that you take the expression before the line and evaluate it at $\alpha = 0$. The notation is also used in other cases than when evaluating derivatives. Usually I would interpret $\frac{\partial f}{\partial x}$ as a function again, and so if I wanted to use a notation similar to your second option I would probably write
$$
\frac{\partial q}{\partial \alpha}(q',t,0).
$$
In my opinion it seems a bit more messy when you put in the $(\alpha = 0)$. So if for example $f(x) = 2x + 1$, I wouldn't write $f(x=1) = 3$ (but some might do that). If you have a function of several variables, like $f(x,y) = xy + 2$, I would write $f(0,y)$ if I wanted to out the variable $x$ equal to zero. Also, I would say that the notation that you have in your comment to your question isn't good since it isn't clear if you are putting $\alpha = 0$ or if it is $q'$ or $t$ that is zero. 
A: Notation with derivatives often get awkward, especially for functions of several variables and derivatives of compositions. Consider expressions like
$$
\frac{\partial f(x,g(x))}{\partial x},\quad \frac{\partial f(x,g(x,y))}{\partial y}.
$$
It gets even messier if you throw evaluations in...
In my understanding, both things you've written down should denote the same thing. Which one you find more convenient is a matter of taste, I think.
