What the meaning of the $dx$ in the expression $u + \frac{\partial u}{\partial x}dx$? What the meaning of the straight handed (d) after the fraction ?

i have a problem i know what the meaning of the partial derivative but i don't know why do we put the dx after the fraction and what does it mean ? 
 A: At least in the places I've run into such use of notation so far, it's used somewhat handwavingly to mean a "really really small displacement", where the displacement in $y$ (or $u$ in this case) is so close to $\frac{\partial u}{\partial x} \mathrm dx$ that one doesn't bother about the difference (the handwavy bit). Or, it's used to mean an 'infinitessimal displacement', smaller than all positive numbers yet not zero. What exactly this means is again not explained.
There may be uses of such notation which are more mathematically rigorous, but in my undergraduate Physics textbooks I've not yet seen them, and something tells me that the book you cited is using the same ideas in this case.
Added later: the two meanings of it end up being interchangeable almost, for in the first the difference between the displacement and $\frac{\partial u}{\partial x} \mathrm dx$ is 'too small to bother', and in the latter it's supposed to be exactly $\frac{\partial u}{\partial x} \mathrm dx$. However, not looking at the (small) difference and saying there is no difference end up with the exact same usage in the end.
A: Short answer:
As explained, $dx$ is the size of the element. The equation will probably look more familiar if you read it as
$$u(x+h)\approx u(x)+u_x'(x)h.$$
A: As depicted in the graph, $dx$ represents the distance between $O$ and $A$. The $d$ means this distance is an infinitesimal, i.e. very small, change along the x-axis. 
Moreover, $\frac{\partial u}{\partial x}$ describes the variation of $u$ in the $x$ direction.
So, the expression : 
$$\frac{\partial u}{\partial x}dx$$
means, you multiply the variation of the function $u$ with respect to $x$ by the infinitesimal change $dx$.
For more information : 


*

*http://mathworld.wolfram.com/Derivative.html

*http://mathworld.wolfram.com/PartialDerivative.html
