The Fundamental Theorem of Calculus states $\int_{a}^{b} f' \ = f(b)-f(a)$
If I define $\frac{df}{dx}:=f'$ and $\int_{a}^{b} f \, dx \ := \int_{a}^{b} f \ $, then I can rewrite above as
$\int_{a}^{b} \frac{df}{dx} \, dx\ = f(b)-f(a)$
Note that I have not defined or assigned any meaning to the terms $df$ or $dx$ individually. If I ignore this important fact and I make the non-rigourous step of simply crossing out the $dx$s, the left hand side becomes
$\int_{a}^{b} df \ $
which looks like the left hand side of the generalized Stokes's theorem, which we know reduces in this case to exactly FTC. So my question is, is there any mathematical meaning to what I did by crossing out the $dx$s? If so, is crossing out the $dx$s "correct" in the context of that meaning? And what actually does the $df$ that I am left with mean?
I am especially curious because it seems that the correct interpretation, especially in the context of the expression looking like Stokes's theorem, is that $df$ is the differential of $f$ (which if I understand correctly is its derivative, although please note I have not learned differential forms). However, we already defined the derivative of $f$ to be $\frac{df}{dx}$, not $df$.
$\newcommand{\bd}{\partial}$The Calculus on Manifolds interpretation of the fundamental theorem of calculus is:
Let $n = 0$. If $a$ and $b$ are real numbers with $a < b$, the closed interval $M = [a, b]$ is an $(n+1)$-manifold with boundary, whose boundary $\bd M = \{b\} - \{a\}$ is the $n$-manifold comprising two points, the left endpoint $a$ with negative orientation and the right endpoint $b$ with positive orientation.
"Integrating" a $0$-form (smooth function) over an oriented $0$-manifold means evaluation multiplied by the sign induced by the orientation.
If $f$ is a smooth $n$-form, i.e., a smooth, real-valued function on $M$ (er, I mean $[a, b]$), then \begin{align*} \int_{a}^{b} f'(x)\, dx &= \int_{[a, b]} df \\ &= \int_{M} df \\ &= \int_{\bd M} f \\ &= \int_{\{b\} - \{a\}} f \\ &= \int_{\{b\}} f - \int_{\{a\}} f \\ &= f(b) - f(a). \end{align*}
This is not a good way to explain the FTC to a calculus student, but it's a fine private realization to have when you study differential forms and integration on manifolds.