so ...
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 ...
http://http://www.physicsforums.com/...p/t-84539.html
... hope it will give you some insight. One must use the term "inverse" with extreme care.
Almost. Strictly speaking, two operations, f and g, are inverse if f(g(x))= x and g(f(x))= x. If you start with , integrate it to get and then differentiate it, you get again. But if you do it the other way around, start with , differentiate it to get and integrate, you get , NOT the same thing.
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 " ".