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.
It's a shorthand: "the derivative of a function, $f$, which is a function of $x$, with respect to $x$."
Your second definition would be
$$\left.\frac{d\left[f(x)\right]}{dx}\right|_{x=x_0}$$
if you take the limit of that expression as $\Delta x \to 0$. Otherwise, it's more of a finite difference than a differential.
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: $${\lim_{dx \rightarrow 0}\frac{(x+dx)^2-x^2}{dx}=}$$ $${\lim_{dx \rightarrow 0}\frac{(x^2+2xdx+dx^2)-x^2}{dx}=}$$ $${\lim_{dx \rightarrow 0}\frac{2xdx+dx^2}{dx}=}$$ $${\lim_{dx \rightarrow 0} \frac {(2x+dx)dx}{dx}=}$$ $${\lim_{dx \rightarrow 0}2x+dx=}$$ $${2x}$$
