Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$? The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is exactly expanded and what its definition is. What if we had equations like follows (even if they do not necessarily make sense):


*

*$\left.\dfrac{d}{dt}\right|_{a=0}f(a+tv)$

*$\left.\dfrac{d}{dt}\right|_{s=0}f(a+tv)$

*$\left.\dfrac{d}{dt}\right|_{t=0}f(a+v)$

*$\left.\dfrac{d}{dt}\right|_{t=0}f(a)$
What are they equivalent to by definition after expansion?
To clarify myself, is the general definition 
$\left.\dfrac{d}{dt}\right|_{s=u}f(a+rv)=\lim \limits_{s\to u}\dfrac{f(a+rv)-f(a)}{t}$?

By definition, 1. is the derivative of $f(tv)$, i.e, $vf^\prime (tv)$. For 2., if $s\neq t$, then the result is $0$. Assuming $v\neq v(t)$ gives $3.$ as $0$, and $4.$ is simply $0$ (it is obvious).
