When not to treat dy/dx as a fraction in single-variable calculus? While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing things like $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$ works out just fine.
So I wanted to know: Are there any particular cases (in single-variable calculus) we have to look out for, where treating $\frac{dy}{dx}$ as a fraction gives incorrect answers, in particular, at an introductory level?
Note: Please provide specific instances and examples where treating $\frac{dy}{dx}$ as a fraction fails

There are places where it is "obvious" that we should not blindly
apply the laws of arithmetic to $\frac{dy}{dx}$ as if it were
a ratio of real numbers $dy$ and $dx$.
An example from another question is
$$
\frac{dy}{dx}+\frac{du}{dv} \overset ?= \frac{dy\,dv+dx\,du}{dx\, dv},
$$
where the left-hand side has a clear interpretation but the right-hand
side does not.
As for any false equation that you might actually be tempted to write
by treating $\frac{dy}{dx}$ as a ratio, however, I have not seen
any actual counterexamples in any of the several related questions
and their answers (including the question already mentioned,
this question, or
this question).
In practice, the problem I see with treating $\frac{dy}{dx}$ as if it were a ratio is not whether an equation is true or not, but how we know
that it is true. For example, if you write
$\frac{dy}{dx} \, \frac{dx}{dt} = \frac{dy}{dt}$
because it seems to you that the $dx$ terms cancel,
without having first learned (or discovered) the chain rule and
having recognized that it justifies this particular equation,
then I would say you're just making an ill-educated guess about this
equation rather than doing mathematics.
(I'll grant that the equation is valid mathematics even if you don't
remember that it's called the "chain rule". I think that particular detail
is mainly important when teaching or when answering questions on
calculus exams that are designed to test whether you were paying
attention when that rule was introduced.)
