F'(x) and F(x) are not inverses, and their sum is not always 0.
... you're crazy?
go see the discussion at this link ...
... hope it will give you some insight. One must use the term "inverse" with extreme care.
The same sort of thing happens with functions: and are "almost" inverses. If you start with x= 2, square it to get 4 and then take the square root, you get 2 again. But if you start with x= -2, square it to get 4 and then take the squareroot, you get 2, not -2. Of course, to solve a quadratic equation, we take the square root anyway but remember that we need that " ".