$dy$ by $dx$ or $dy$ divided by $dx$ I was always taught do not say "$dy$ divided by $dx$", instead "$dy$ by $dx$" because it's not really dividing. 
I then studied differentiation from first principles, where one takes two points on a curve: eg. $(x, y)$ and $(x+\delta x, y+\delta y)$
$$\therefore \text{ gradient} = \frac{(y+\delta y)- y}{(x+\delta x) - x}$$
I'll skip the continuation but one gets the derivative of the equation through some algebra. If you simply expand the brackets instantly you simplify it to $\frac{\delta y}{\delta x}$ which is a division.
Firstly, why is the sign different - is it just because it is easier to right $d$ than $\delta$?
Secondly, why can one not say $dy$ divided by $dx$?
Thanks
 A: One can certainly say it.  The question is whether $dy$ and $dx$ actually mean anything by themselves, such that $dy/dx$ is their quotient.  And the answer is yes: they refer to the $x$ and $y$ coordinates of a displacement along the tangent line to the curve.  
A: I don't think there's any harm in saying "$dy$ divided by $dx$".  But${}\ \ldots$
I was taught that
$$
\frac{dy}{dx} = \lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x}.
$$
Certainly $\Delta y$ and $\Delta x$ are numbers.
The differentials $dy$ and $dx$ are thought of as infinitely small but nonzero increments of $y$ and $x$.  This raises a question of whether one can really make sense of such a concept as infinitely small but nonzero quantities, and whether one can then do calculus with them, using them in just the way in which the notations $dy$ and $dx$ are used.  Robinson's nonstandard analysis does make use of infinitely small quantities, but not in just exactly the way in which Leibniz (who introduced the notation) and Euler (who used it more extensively than anyone else) used them.
The whole theory of calculus can be justified without proving that infinitely small increments can be taken literally.  That justification was done in the 19th century by "epsilons and deltas".
So the fact that one doesn't take them literally is why some people say you shouldn't speak of dividing when it's not clear that it's really dividing.
However, speaking of infinitely small increments is an immensely useful heuristic.
A: It is helpful, particularly once you start studying multivariable calculus, to recognize the derivative $\frac{dy}{dx}$ as not a real number but a linear map from tangent vectors on the domain to tangent vectors on the image.
From this, some misguided people get the impression that one "shouldn't" call $\frac{dy}{dx}$ in ordinary one-variable calculus a ratio. At a given value of $x$, tangent vectors $dx$ at $x$ and their pushforwards $dy$ can be canonically identified with real numbers, and the derivative of $y$ at $x$ is 100% rigorously the quotient $\frac{dy(dx)}{dx}$ (when $dx\neq 0$), which one can show does not depend on $dx$. As Robert points out, this has a very helpful and still 100% rigorous geometric interpretation when you intepret $y$ as a graph, and $dx$ and $dy$ as  infinitesimal horizontal and vertical displacements.
