A bird in the hand is worth three in the bush. If I could give someone from a non-mathematician background, e.g. programmer, who sincerely wants to get a peek behind the curtain and know what calculus really IS without diluting anything in an unacceptable way I'd show them I. Gelfand et al "Sequences, Combinations, Limits". Once you have the idea of a limit of a sequence down it isn't hard to learn the limit of a function (epsilon, delta) and move on from there. A good second step is Apostol's calculus vol. 1 just the first chapter where he calculates an integral from scratch with no "fluff" (details you don't need just yet). (Yes, learning integration first actually makes more sense than learning derivates.) Once you do that you can continue with Apostol (highly recommended.) or use Spivak's calculus (tougher but even more highly recommended).

To wrap up Gelfand is one bird in the hand, Apostol chapter 1? with integration from stratch is another bird in the hand. That would be a good first two steps to get a sense of what calculus is really about.