TITLE: How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
QUESTION [96 upvotes]: I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
There is one strong argument against this: We tell students that $dy$ and $dx$ mean "a really small change in $y$" and "a really small change in $x$", respectively, but these notions aren't at all rigorous, and until you start talking about nonstandard analysis or cotangent bundles, the symbols $dy$ and $dx$ don't actually mean anything.
But it gives the right intuition! For example, the Chain Rule says $\frac{dy}{du} \cdot \frac{du}{dx}$ (under appropriate conditions), and it looks like you just "cancel the $du$". You can't literally do this, but it is this intuition that one turns into a proof, and indeed if one assumes that $\frac{du}{dx} \neq 0$ this intuition gets you pretty close.
The debate about how rigorous to be when teaching calculus is old, and I want to steer clear of it. But this leaves an honest mathematical question: Is treating $\frac{dy}{dx}$ as a fraction the road to perdition, for reasons beyond the above, and which have not occurred to me?For example, what (if any) false statements and wrong formulas will it lead to?
(Note: Please don't worry, I have no intention of telling students that $\frac{dy}{dx}$ is a fraction; only, perhaps, that it can usually be treated as one.)

REPLY [6 votes]: A note from a publication (my own) that occurred several years after this question was asked.  $\frac{dy}{dx}$ can be considered a fraction of differentials.
You can think of differentials as infinitesimal values that are related to each other.  Non-standard analysis showed that although 19th century mathematics viewed infinitesimals as problematic, they can be easily treated as ordinary mathematical objects, capable of division, multiplication, etc.
There is no problem treating $\frac{dy}{dx}$ as a fraction, but there is a problem in higher-order derivatives and differentials, but that is because we are using a notation that doesn't support it.  If you take the idea of $\frac{dy}{dx}$ being a fraction seriously, then, to find the second derivative, you are taking the derivative of a fraction.  Therefore, you have to apply the quotient rule.  If you apply the quotient rule to $\frac{dy}{dx}$ you do not get the typical result of $\frac{d^2y}{dx^2}$.  Instead, you get:
$$\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$$
Or, written less ambiguously:
$$\frac{d(d(y))}{(d(x))^2} - \frac{d(y)}{d(x)}\frac{d(d(x))}{(d(x))^2}$$
When written this way, the second derivative can be considered actual fractions just like the first derivative.  Third and higher derivatives are even uglier, because you are taking the derivative of that.
You can see more details of this in "Extending the Algebraic Manipulability of Differentials", Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 26(3):217-230, 2019.  And, if anyone is concerned for its validity, it had a further review in Mathematics Magazine 92(5), pp. 396–397 in their "Reviews" section.