[noblethrasher]

Doing the exercises is not rewriting calculus, but it does lead to the discovery of useful pseudo-theorems about mathematical expressions and calculus-based techniques for manipulating them.

An actual rewriting of calculus would involve discovering proofs for things like the Mean-Value Theorem or the Fundamental Theorem of Calculus. Short of that, you could wrestle with trying to discover proofs for theorems so that you can at least appreciate the canonical ones. Incidentally, this is what every mathematics major has to do in order to earn their degree.