Here's a silly-but-perhaps-provocative way to talk about calculus.
The only atom to start with is +1+1+1+1+1+1+1+… ↣ 5+5+5+5+5+5+5+…. Some people call
this "to quintuple" but I'll just call it ƒ. Repeated addition of the
number 5. I'm going to build up large classes of functions from this,
using "meta-moves" i.e. just operating on the function.
The first meta-move I'll allow myself is to stick the +5+5+5+5…
anywhere I want on the number line. Tangent picking. Also known as b
in y=mx+b.
Comovement. I'll also allow myself to jiggle the 5's. Sometimes I'll
be talking about
+5+5+6+5+5+... and sometimes I'll Shift operator before Co-moving . If I did that a few times I might have +5+5+7+6+5+3+5+5+5+….
Recursion. multiple derivatives. ƒ−−1 = +5+4+3+2+1+0−1−2−…. Smooth
functions are the ones that I can write some complicated difference
pattern "at the beginning" with a (potentially infinitely long) string
of ƒ−−1+++½++++0++++++0+++++++14….
(Remark about digits. The ++ digit, +++ digit, ++++ digit, +++++ digit can be encoded as a polynomial….)