A derivative is calculus. I'm not sure how you expect to arrive at an answer couched in calculus terms without using any calculus - by the time you reach the final step of the proof, you've necessarily used calculus.
Other than that, I can think of two solutions that "sort of" don't use calculus. One involves taking a limit, and the other involves the fact that an integral is the inverse of a derivative. Are either or both of those considered "using calculus"?
re(2): Err....
