Great overview.
1. Derivatives
2. Integrals
3. Applications — function approximation; solids of rotation; vectors; etc.
4. Multivariate — partial derivatives, multiple integrals, etc.
Glancing at a nearby community college’s course catalog, they have similar split.
We covered all the proofs in real analysis 1 (derivatives; sequences) and 2 (integrals; measure).
This textbook (work in progress but mostly complete) is an attempt to do exactly that: start at foundational math like group theory and topology and build up to higher and higher levels of math.
I worked through the first few chapters a while ago, and it's very good. My only issue is that sometimes he assumes knowledge of things that I didn't know about, but cursory googling was good enough for those situations.