Why is the notation for differentiation like this? Consider the notation for denoting the differentiation of a function $f(x)$. 
$$\frac{d[f(x)]}{dx}$$
I mean, this notation doesn't make any sense. $dx$ means a vanishingly small $x$, which can be understood, if we take $x$ to mean $\Delta x$ in a loose sense. But what does $df(x)$ denote? 
I mean, ideally, shouldn't differentiation be denoted as
$$\frac{f(x_0 + d\Delta x) - f(x_0)}{d\Delta x}$$
Is this done just to simplify things, or is there another reason? 

The $d[f(x)]$ indicates the vanishingly small change in $f(x)$ corresponding to the vanishingly small change in $x$ indicated by $dx$. Their ratio is the derivative.
