When is it legal to use $dx$? I still have some trouble grasping the true mathematical meaning of $dx$. I have the intuitive understanding that it is an "infinitely tiny amount of $x$" but I don't know in what cases I can use it.
Here is an example : 
is $\lim_{x\to 0} \frac{2x + 32x^2}{e^x}$ equivalent to $\frac{2.dx + 32.dx^2}{e^{dx}}$ ?
Thanks a lot
 A: Let's start with your $f'(x)= \lim\limits_{\Delta x\to 0} \frac{\Delta (f(x))}{\Delta x} = 2x$ when $f(x)=x^2$
Sometimes the derivative $f'(x)$ is written $\frac{df}{dx}$ or $\frac{d}{dx} f(x)$.  All three of these are simply notation for $\lim\limits_{y \to x} \frac{f(y)-f(x)}{y-x}$ or $\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ and some people describe this as the limit of the ratio of the change of $f(x)$ to the change in $x$ leading to your $\lim\limits_{\Delta x\to 0} \frac{\Delta (f(x))}{\Delta x}$ or $\lim\limits_{\Delta x\to 0} \frac{\Delta f}{\Delta x}$: the notation $\frac{df}{dx}$ is suggestive of this limit but the $dx$ does mean anything on its own here       
Since the derivative operation can be applied to different functions, further notation lets $\frac{d}{dx}$ represent a derivative operator. It can be applied more than once: for example a double derivative would involve a double operation and is sometimes written $f''(x)= \frac{d^2}{dx^2}f(x)=\frac{d^2f}{dx^2}$; with $f(x)=x^2$ you would get $\frac{d^2f}{dx^2} = 2$.   Again this is notation, and the $dx$ is part of the notation without having a stand-alone meaning; in particular it is not being squared
You will also see $dx$ as part of the notation for integrals: for example $\int_6^9 x^2\, dx = 171$.  But while this might be suggestive of a sum (the long s in the integral sign) of lots of pieces for $f(x)$ with widths represented by $dx$, it is again notation and the $dx$ does not have an independent meaning here, just being part of a wider expression for the integral operation
Some people write expressions such as $d(fg)=f\, dg + g\, df$ for the product rule.  Typically this is shorthand for $\frac{d}{dx}(fg)=f \frac{d}{dx}g + g \frac{d}{dx}f$ and is another example of the use of notation
There is nothing to prevent people from giving a particular meaning to a bare $dx$ so long as they define it in a meaningful way.  Then again it would still be notation, with that definition      
A: The differential being an infinitely tiny amount is a misconception. In modern setting, the differential is a linear map, specifically the linear part of the variation of a function when you vary the variable. In other words, when a function is differentiable there is a constant $c$ such that
$$f(x+dx)=f(x)+c\cdot dx+o(dx)$$ ($o(dx)$ is a sublinear term as a function of $dx$)
 and by definition $$df(x)=c\cdot dx.$$
The constant $c$ is well know to be the derivative of $f$, and $dx$ can be an arbitrarily large increment.
Actually, differentials have little to do with limits.

On the picture, the function is in blue. Its differential (at $x=1$)  in magenta and the remainder in black.

