I was all set to begin Calculus 2 when I thought, "I should have a more intuitive sense of what's happening with differentials before I move on."
I want to tell you what I've learned and ask you all to help me put it together more cohesively than I have done. I'm just going to list what I know. Please point out my inconsistencies. And please forgive me if the following seems to ramble, I'm just sharing my thought process and I want it to have more rigour.
I've seen Qiaochu Yuan saying a few times that $dy$ and $dx$ are not numbers. I didn't know what he meant. So, I read elsewhere, and learned about infinitesimals. I read that an infinitesimal is a concept similar to zero, but not. It is a small, unmeasurable value. That there is no value you can multiply an infinitesimal by to get any real number. Yet, it's not zero. I read that this is an important distinction.
So, when I read $\frac{dy}{dx}$, this is not a fraction: it should be read as one symbol, and interpreted as shorthand for $\lim\limits_{\delta x \to 0}\frac{\delta y}{\delta x}$.
A problem I still have, though, and it's surprising even to me that I don't get this yet, is this: if I'm finding the gradient of a curve at a point, then I'm finding the gradient of the tangent line at that point. I do this by setting up the quotient $\frac{\delta y}{\delta x}$ where both $\delta y$ and $\delta x$ are measureable values, real numbers. as soon as I write $\lim\limits_{\delta x \to 0}$ in front of the quotient, I'm saying that the quotient coupled with the limit is not a quotient. So $\delta y$ and $\delta x$ are still real numbers, but $\frac{dy}{dx}$ is a symbol that means the tangent, the limit, or $f'(x)$.
So two things I'm not sure of at this point: it seems that if $\frac{dy}{dx}$ is a symbol in its own right, then $dy$ and $dx$ are not independent symbols. And if they are, I don't know what they mean. I will guess that $dy = \lim\limits_{\delta x \to 0}\delta y$ and $dx = \lim\limits_{\delta x \to 0}\delta x$. To say that $dx$ is infinitesimally small is to say that it is infinitely close to zero but not zero.
But, if two points are infinitely close to one another, where the distance between them is an infinitesimal, they are still two points aren't they? They can only be the same point if the distance between them is zero, and an infinitesimal is not zero. And so, how can I find the gradient of the tangent line when the tangent is really a secant?
I know this is important to understand, and at some point I thought I understood it intuitively. But my understanding has slipped and I'm here. I can tell myself "it just is" but it's not much of an argument and it's not satisfying. I want to understand everything as much as I'm capable of doing so, and take nothing on faith.
Once I understand this, I will be able to write the chain rule without multiplying by infinitesimals, then the same with the substitution rule, then I move onto Calculus 2. Until I understand this, I can't move on.
I'm guessing my problem comes from a lack of rigor in some textbooks, and that makes my overall understanding weakened. If I try to solve my own problem, I might say that $\frac{dy}{dx}$ is not, in fact, the gradient of the tangent at all. It is the gradient of the secant, but it is the value that is infinitely close to the tangent. It "approaches" the tangent. But it still comes down to this: if two values are not the same, and their difference is infinitely small, what does that mean? Sometimes they are considered the same, sometimes they are not. Which is it? If something is infinitely small, it has a value greater than zero, but it's not zero, it can't accumulate to become any real number.
It's not a number. It's a concept with the above properties and that's all there is to it. Is that the answer?
I understand the idea is to be able to get infinitely close to a value while avoiding a division by zero. And knowing that is almost enough for me to move on. But I think there's more for me to understand here, and a lot rests on it.
Thanks for reading this far, if you did. I appreciate your feedback.
Also: another question. If I have $dx$ and $du$ and they are both infinitesimals, then $1+dx = 1+du$, right? Yet, $du$ and $dx$ are not the same, right? In an integral, I can't just switch out a $dx$ for a $du$, yet they both represent the same concept.