[tzs]

I don't recall Bayes' theorem involving calculus. Are you sure you aren't thinking of some other theorem?

Bayes' theorem follows straightforwardly from P(A & B) = P(A|B) P(B) and P(A & B) = P(B & A). The latter tells us that we can swap A and B in the former without changing the value, giving us P(A|B) P(B) = P(B|A) P(A).

Rearranging gives P(A|B) = P(B|A) P(A) / P(B), which is Bayes' theorem.