\section{Formal calculus}
Here's an idea:
introduce the derivative (and the anti-derivative, aka `integral')
formally. That is, just tell people that
$[x^n]'=nx^{n-1}$
(linearity, product rule, chain rule, $\sin/\cos/\exp/\log$)
and then work with them for a bit -- only \emph{afterwards} explain what they
are and prove things.
That is:
- students won't care/follow proofs the first time around;
  they won't even remember/understand definitions.
- once you've worked with something, then you are motivated to understand it.
...think of my experience with Spectral Sequences:
they were lots of fun to work with, and only later did I understand why they
converged etc.

...then once people can do calculations
(including differentiation and integration in several variables),
only then explain what's happening and prove things.

The idea is that this breaks it up into bite-size chunks.
Further, it makes them more like ``mathematicians'' approaching calculus: they
can do the calculations, so they can learn the reasoning.

(if you try to teach calculations and reasoning at the same time, they'll just
learn the calculations and tell you to shut up with the philosophy)
