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? 

You are close. What you are talking about is the difference quotient.$${\lim_{dx \rightarrow 0} \frac {f(x+dx)-f(x)}{dx}=\lim_{dx\rightarrow0 }\frac {\text{change in }f(x)}{\text{change in }x}}$$ . So using${f(x)=x^2}$  as an example: $${\lim_{dx \rightarrow 0}\frac{(x+dx)^2-x^2}{dx}=}$$ $${\lim_{dx \rightarrow 0}\frac{(x^2+2xdx+dx^2)-x^2}{dx}=}$$ $${\lim_{dx \rightarrow 0}\frac{2xdx+dx^2}{dx}=}$$ $${\lim_{dx \rightarrow 0} \frac {(2x+dx)dx}{dx}=}$$ $${\lim_{dx \rightarrow 0}2x+dx=}$$ $${2x}$$ 
