What is the ultimate example of physicists point of view vs mathematicians point of view?

What is the ultimate example of a physical point of view versus a mathematicians point of view?

The following example illustrates a problem between physicists and mathematicians that has been the subject of conflicting approaches as theoretical physics.

physical point of view versus a mathematician's point of view

Consider the discriminant equation:

df(x) dx = f(x).
I would agree that the correct step is to divide both sides of this equation by f(x).
df(x) dx1f(x) = 1.
(In the special case where f(x) = 0 is properly excluded, this is a trivial solution).
impressions
df (x) dx
Both the geometry and the difference are well defined with limited operations close to zero, and there is a correct real-world relationship between the two. The closer to zero, the closer the quotient is to a certain value, so that certain value is the effective value of that expression. (The definition of mathematics is certainly more accurate than that.)
But the physicist will now multiply both sides of this equation by dx without any hassle. The result is simple algebra.
df(x) f(x) = dx,
But except in physics it makes no sense. Because there is no more good part where numbers and rulers contribute meaningfully. The X in DF(X) does not have the same meaning as the X in Dominator, and the meaning is not clear. Oh well, let's forget the left hand X because the physicist thinks it doesn't make sense anyway.
dff = dx

Then merge the two sides to find it:

log(f(x)) = x + C
In fact, my discriminant equation professor (mathematician) said that any method for solving a differential equation is fine. Because someone turns the solution into a real equation to see if it works. But the physicist wouldn't mind this step. He will be very happy with the above conclusion.
As an example of the same word treatment for derivatives, the following is directly quoted from Wikipedia:
... an infinite element of a three-dimensional velocity space d3v, whose intensity is centered around the vector of velocity v and f(v) is d3v.
f (v) d3v = (m2πkT) 3 / 2e - mv22kT d3v, ....
Consider the reference to the "infinite element", a concept familiar to Newton and still the same to physicists today, but not explained in modern mathematics.
A related example is the Drake delta function, traditionally interpreted as oted(x), a concept loved by physicists but not recognized by mathematicians. its definition is
δ(x) = 0 ∀ x ≠ 0,
∫ - abδ(x) dx = 1 ∀ a, b > 0.
Mathematicians are not aware of this concept. Because it violates the reasonable definition of integral. But physicists like it because they usually struggle with high multiple integration. Dirac delta functions appear in the fog and POP! Two levels of integration disappear.
Addendum: Annotations show that mathematicians have expressed strictly and precisely the concepts described here. However, physicists do not use these agents. I usually draw small boxes with arrows moving in and out to show the difference.
x.