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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 $lim_{xto 0} frac{2x + 32x^2}{e^x}$ equivalent to $frac{2.dx + 32.dx^2}{e^{dx}}$ ?

Thanks a lot

Let's start with your $f'(x)= limlimits_{Delta xto 0} frac{Delta (f(x))}{Delta x} = 2x$ 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 $limlimits_{y to x} frac{f(y)-f(x)}{y-x}$ or $limlimits_{h to 0} frac{f(x+h)-f(x)}{h}$ 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 $limlimits_{Delta xto 0} frac{Delta (f(x))}{Delta x}$ or $limlimits_{Delta xto 0} frac{Delta f}{Delta x}$: 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

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

$$f(x+dx)=f(x)+ccdot dx+o(dx)$$ ($o(dx)$ is a sublinear term as a function of $dx$)
and by definition $$df(x)=ccdot dx.$$

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.