The essential rules of differentiation are:
- linearity
- Leibniz etc.
  (product, power, inverse/quotient)
- chain rule

that is, it's how differentiation interacts with the basic
operations on functions:
- addition/scalar
- mult/division
- composition

there's lots of ways of seeing these:
- geometric
  yay! intuition!
- graphs
  esp. from secants on graphs
  these are a -good- way of seeing 'em,
  as they provide a global/local picture when the infintesimal
  one is a bit obscure
- from definition
  easy to write down,
  but -dry- as -sin- unless you give geometric intuition;
  these should -really- be seen as -formalizations-
  of the geometric intuition
- algebraic
  product is immediate from 2-variable differentiation;
  power, inverse/quotient follow -formally- from product;
  this is important b/c it means that we can axiomatize
  a derivation as something linear that satisfies the product rule,
  and everything else follows

  quotient also follows from inverse + product + chain rule,
  which is how I understand it (er, as why it's so ugly).
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- geometric
  -linear: pretty obvious; concatenate/scale rods
  -product: scaling rectangle
  -power: scaling cube: binomial -counts- the edges,
     but all you need is that the edges are smaller
  -inverse/quotient:
    there -is- a geometric interpretation in terms of:
    "a rectangle with area 1, and sides f, 1/f: how does 1/f change
     to keep area constant"
    ...and this is the natural visualization of the algebra involved,
    but I find the algebra clearer. it does work ok, esp. for
    explaining the negative, but it's not great...

- graphs
  -linear: really obvious from graphs;
   scaling/adding just affect rise, in exactly the right way
  -product: cute: do in 2D, and it's like the picture for Lie bracket
  -power: write out numbers for secant slopes, and it's pretty easy
    (this is a bigger version of the binomial rule,
     and that's how it should be understood)
    this also works for exponents and logs
  -inverse
    this one is quite good:
    the line from the origin to (x,1/x) has the opposite slope/
    is the mirror reflection of the tangent line,
    hence tangent line takes x steps to go down 1/x,
    hence has slope $-(1/x)/x=-1/x^2$
  -quotient:
    you're kidding, right?

- definition
  easy; only tricks are:
  - binomial rule for power
  - multiply through for inverse/quotient rule

- algebra
  trivial
  it's worth noting that (in addition to the quotient/inverse rule)
  the fact that -constants- have derivative 0 also follows from
  the product rule:
  f' = (1.f)' = 1'.f + 1.f'
  So 1'.f = 0 for all f, including f=1, so 1'=0.
    (kinda similar to proof that group homo takes id to id)
  ..so by linearity, all -constants- have derivative 0.
  What's a constant? it's any scalar -- it's what differentiation
  is linear w/r/t.
  (so of course we define a derivation to be linear w/r/t a subring,
   since we define derivatives on rings)
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applying/remembering:
- linear is easy/obvious,
  but remember to always take constants out,
  and -recognize- constants (EG, pi, y (when multi-variable diff))
- product is easy, power is v.easy
- quotient: just memorize (lotta mneumonics)
- chain: this takes some drilling to get right when doing
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? how does differentiation relate to convolution?
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NB: a -ring- axiomatizes addition/mult;
just composition gets axiomatized as a semigroup/monoid
...and differential algebra is easy

OTOH, axiomatizing -all- of these, including composition,
is pretty tricky;
is it just "End A" for some algebra?
how to axiomatize that?

(...and if you also want to do convolution???)
