I recently read the book Calculus Made Easy. The way it teaches the material is very intuitive and if you understand the premises you should be able to re-derive all of the formulas yourself. I would highly recommend it to high school students and teachers alike.

I remember being taught about derivatives with the formal limit formula, then all the tricks for finding them (power rule etc), and finally finding function extrema using derivatives; all before developing much intuition about the concept.