to put it in a different way: equivalent is not equal. we "lump stuff together" when we make equivalences.
you can define f to be equivalent to g if f'(x) = g'(x), that is a perfectly good equivalence. but it is NOT equality.
y = f(x) = 2 and y = g(x) = 0 have the same derivative, and in some sense "act the same". but clearly 2 isn't 0.
this is why we put the +C after integrals, because we don't get a unique function, we get an equivalence class of functions.
Suppose is a known function.
Is it then correct to write: .
I mean, is true.
And if and only if is true, then so is , right?
I would drop the "if and only if".
The converse is not always true.
. . If , then may be equal to
The derivatives are equal, but the functions are not equal.