The subtlest point about the derivative is:
- the derivative of a function is a -co-tangent vector field
(that's what the dx is for)
...and the internal derivation operator
$D\from \cD \to \cF$
is b/c we have a canonical trivialization,
due to the covector field $\dx$:
you're taking a ratio.

This is -much- easier to see in multiple dimensions
(1 dimension is too small),
as the derivative of a functional is -not- a functional.
[This is easier to see dually, with gradient vector fields:
 the gradient is a -vector field-, not a -function-]
This is simply reflecting the fact that the cotangent bundle has more
than one dimension.

This seems to become trivial when you then say:
"Ok, so the derivative of a functional is a 1-cycle.
 A 1-cycle is something that you can integrate over 1-cells.
 How?
 By pulling it back to the interval and evaluating it there."
This idiotically trivial sounding answer is actually reflecting the
fact that the higher dimensional stuff is what follows -naturally-
from the 1-dimensional definition.

[The Fundamental Theorem becomes almost a -definition- in this context.]

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There is a very natural way of seeing this though,
which is the intuitive description of df
(the total derivative):
- in one dimension, asking
  "what is the slope?"
  is unambiguous, as we always go -right-
  (actually, if you go left, the slope is the negative)
- in several dimensions,
  you instead must ask:
  "what is the slope -in some direction-?"
  If you like, you can ask this for the cardinal directions:
  those are the -partial- derivatives.

  More precisely, the total derivative answers the question -in every
  direction-
  (it's confusing at first b/c it's a functional on $T\bR^n$:
   directions are -vectors-, not -angles-,
   so there's a confusing normalization to get slope).
  ...and that's exactly what a covector field is:
  a value -in every direction-.
  (I understood this at some point in my 1st year at Harvard,
   and conveyed it -in some sense- to my 190s students,
   but I only just wrote it down properly)
