== Notation 1 ==

In the integral calculus, one encounters a notation such as $\left.F(x)\right|^a_b$.  This is what we mean:
$$ \int_a^b dF = F(b) - F(a) \equiv \left.F(x)\right|^a_b. $$

== Notation 2 ==

The notation $\left.F(x)\right|_a$ is a quite different thing.  It just means the fucntion $F(x)$ evaluated at $x = a$.

So, note the following.

$$ \left.\frac{\partial D}{\partial x}\right|_x \equiv \frac{\partial D(x,t)}{\partial x},\;\; \left.\frac{\partial D}{\partial x}\right|_{x+\Delta x} \equiv \frac{\partial D(x+\Delta x,t)}{\partial (x+\Delta x)}. $$

== Bonus ==

While we are at this sort of stuff, note that

$$ \left.\frac{\partial D}{\partial x}\right|_{x+\Delta x} - \left.\frac{\partial D}{\partial x}\right|_{x} = \frac{\partial^2 D}{\partial x^2} \Delta x$$

if $\Delta x$ is taken to be the infinitesimal (i.e., if it can be considered as the mathemtical zero).

Note that there is no zero in physics, i.e., in reality&mdash;there is only an unresolvably small quantity<<fn(One quite mundane, but very true, reason is because increasing the resolving power is always a very extremely expensive proposition, if you are at the bleeding edge of pushing beyond the state of the art.  More fundamentally though, the following is to be noted.  Often, when one succeeds in pushing the resolution, one discovers new physics.  In such a case, the thing that we call zero in mathematics is dependent on which physics we are talking about.  For instance, the mathematical zero for Newtonian mechanics is in fact the mathematical infinity for quantum mechanics!  The &ldquo;particle string&rdquo; has a zero size in Newtonian mechanics, but it has an infinite size in quantum mechanics, which is concerned with infinitely smaller &ldquo;nano springs,&rdquo; in the language of Homework 3.1.)>>