Tracy wrote:
> the chain rule is given by the following expression:
> deriv (f . g) x = deriv f (g x) . deriv g x
Oleg corrected the blog author:
> Incorrect form: the first "." is composition, the second
> should be multiplication. The composition on the right
> side should be between f' and first g.
To clarify using J notation:
chain =: 2 : 'u@:v D.1 -: u D.1 @: v +/ .* v D.1'
The conjunction chain takes two verb (function) arguments and produces a
tautology (i.e. a verb that produces 1 for any input). At least in theory.
1&o. chain (3 2&p.) _78 0.9 1p1 1j5
1
-Dan
PS: Using J4 notation, before the trimming of Section II.F:
chain =: @:D.1 -: ([.D.1 @:]. (+/ .*) (].D.1))
I'm particularly enamored of the @:D.1 part. This is pure definition: the
first derivative of the composition. The function arguments have been
abstracted away. This is analogous to how tacit verbs abstract away noun
arguments (e.g. +/%# means "the sum divided by the tally"; no arguments are
mentioned).
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