I am following N. Piskunov's book for Integral calculus. The indefinite integral is introduced in the notation $\int f(x) dx$ without explaining the meaning of dx. At first I thought it was mere notation to indicate what variable we are integrating in reference to.
But when attempting integration by substitution, we can clearly treat $dx$ as a a part of the expression, and not just notation. For example, $$\int f'(x)/f(x) dx $$ and $$t=f(x)$$ then $$dt/dx = f'(x)$$ and then we can substitute in dx to get the answer and it works.
Why can we treat dx like this in order to get results? Is it not only supposed to indicate the reference variable of integration.
Edit: I realised my question was too vague. I understand how the differential operator works and that dx, dy etc can be used algebraically. But what I'm confused about is what the expression $f(x)dx$ means if $dx$ does not just notation. And then the next obvious question is what the integral operator $\int$ doing to it such that it results in the antiderivative of $f(x).$
A definite integral can be defined as the limit of a Riemann sum.
$$\int_a^b f(x) ~dx = \lim_{\Delta x \to 0} \sum_{k=0}^{N - 1} f(x_k) \Delta x$$
where $N = \lfloor \frac{b-a}{\Delta x} \rfloor$ is the number of sub-intervals, and $x_k \in [ a + k\Delta x, a + (k + 1)\Delta x]$ is arbitrarily selected within each sub-interval.
The summand represents the area of a rectangle, with a height ($y$) of $f(x_k)$ and a width of $\Delta x$.
Notationally, you can think of $\int$ as a replacement for $\sum$, and $dx$ as a replacement for $\Delta x$. Just spelling variants of the letters S (for sum) and D (for difference).
A similar argument applies to the “fraction” notation for derivatives.
$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y := f(x + \Delta x) - f(x)}{\Delta x}$$
As $\Delta x$ and $\Delta y$ approach 0, we call them $dx$ and $dy$.