Connection between dx and dy as component functions and infinitesimals

Question

I am watching an elementary video series on differential forms by Michael Penn. In this video, he describes the definition of $dx$ and $dy$ as some component functions for the coordinate system of a tangent space, where $dx: T_PC \rightarrow \mathbb R$ and $dy: T_PC \rightarrow \mathbb R$, which respectively give the first and second components of a vector $v$ in a tangent space $T_PC$ at a point $P$ on a curve $C$ in $\mathbb R^2$.

He goes on to show that if a curve $C$ is defined by $y = f(x)$, then we can show that the quotient of the two component functions is equivalent to the derivative of $y$ at some point $P$, i.e. $\frac{dy}{dx} = f'(a)$, where $(x, y)P = (a, b)$.

From my understanding (and from this video where he talks of whether or not the quantity $\frac{dy}{dx}$ can be called a fraction), this notion of the ratio of the two component functions is supposed to formalise the definition of Leibniz's notation of a derivative.

What I don't understand is what the connection is between $dy$ and $dx$ as component functions of the vectors in a tangent space and $dy$ and $dx$ as infinitesimal changes in $y$ and $x$. I understand that an "infinitesimal change" is probably not well-defined in standard analysis, but I'd still like to build some intuition and understanding behind why these two seemingly different definitions of $dy$ and $dx$ are actually connected.

Note: I understand that this question has been asked here, but I didn't really understand the answer. It seems to me that both definitions of $dy$ and $dx$ as component functions and as changes are defined, but the connection between them isn't really explored. Please correct me if I'm wrong in thinking this.

Answer 1

"It seems to me that both definitions of $dy$ and $dx$ as component functions and as changes are defined"

This is your basic mistake. In the traditional setting, $x$ is the dependent variable and $y=f(x)$ is the independent variable. Accordingly, $dx$ will indeed represent an (infinitesimal) change in $x$, but $dy$ is different from the corresponding change in $y$. To elaborate, one starts with an infinitesimal change $\Delta x$ in $x$ and computes the corresponding change in $y$ as $\Delta y = f(x+\Delta x)-f(x)$. What happens is that, if $f$ is differentiable, then defining $dy=f'(x)dx$, we observe that $dy$ and $\Delta y$ are "almost indistinguishable" in the sense that the ratio $\frac{\Delta y}{dy}$ is infinitely close to $1$. In this sense (and only in this sense) one can think of $dy$ is the (infinitesimal) change in $y$.

"I understand that an 'infinitesimal change' is probably not well-defined in standard analysis"

You should understand, instead, that such claims (sometimes found on MSE and elsewhere) are merely a *bad pun. If the term standard is taken in its generic sense of "traditional", then the statement is circular/vacuous, to the extent that in post-Weierstrassian mathematics one traditionally avoids teaching infinitesimals. On the other hand, if the term is meant as a reference to the technical meaning in "nonstandard analysis", namely the theory developed by Abraham Robinson to do infinitesimal analysis rigorously in the context of classical logic and and classical set theory, then the claim is patently false: infinitesimals are well-defined in classical mathematics namely nonstandard analysis.