TITLE: How is it that treating Leibniz notation as a fraction is fundamentally incorrect but at the same time useful?
QUESTION [19 upvotes]: I have long struggled with the idea of Leibniz notation and the way it is used, especially in integration.
These threads discuss why treating Leibniz notation as a fraction and cancelling differentials is incorrect, but also go on to say that the notation is suggestive and we use it because it simplifies things:
What is the practical difference between a differential and a derivative?
If dy/dt * dt doesn't cancel, then what do you call it?
In them they say to treat the differential at the end of an integration expression as a "right parenthesis". This throws me off a bit because we can so easily do something like:
$$\int\cos(3x) \, dx\\
u=3x\\[2ex]
du=3\,dx\\[2ex]
\frac{1}{3}du=dx$$
and then proceed to integrate:
$$\frac{1}{3}\int\cos(u) \, du$$
and arrive at the correct answer with "incorrect" notation. I am supposed to treat the differential as a parenthesis but using this notation the differential seems to have a value.
How does this incorrect notation do such a good job ensuring that we do not disobey the "reverse chain rule" and ensures that our integrand is in the form $f'(g(x))\,g'(x)$ ?
People often say that it is very suggestive and I am wondering how. Excuse the LaTeX if it looks weird. This is my first time using it.

REPLY [2 votes]: Leibniz notation is useful and suggestive because it is fundamentally correct.  Anyone who tells you otherwise simply doesn't know what they're talking about.
It's understandable how this misconception came to be.  When Calculus was first developed in the 17th century, it was clear that it didn't have the same level of logical rigour that arithmetic and geometry had had for about 2000 years prior.  It wasn't until the 19th century that this changed, and the rigorous development at that time did not include differentials or any other infinitesimal quantity.  So at one time, it was fair to say that we did not have a fundamentally correct account of Leibniz notation.  But it was never correct to say that the notation itself was fundamentally incorrect, only that if it was correct, then we did not know how.
Since another hundred years or so has passed since, it should be no surprise that the situation has changed.  If you can derive a contradiction from a notation, then that's a sign that it is fundamentally incorrect, but it's also possible that one can figure out rules on its usage that will prevent contradictions.  And if a notation is useful and suggestive, then that's a sign that it is fundamentally correct, at least when appropriately used.  So there was never a time that anybody could have reasonably believed that they knew that Leibniz notation was fundamentally incorrect, and indeed, we now know that it is fundamentally correct.