All the ideas are from Jerzy Karczmarczuk's work, see http://users.info.unicaen.fr/~karczma/arpap/diffalg.pdf.
A simple example, if we define
foo x = x*x
then the function
foo' = deriv foo
will behave as if its body was 2*x.
Dif type is the type of differentiable numbers.
It's an instance of all the usual numeric classes.
The computed derivative of a function is is correct
except where the function is discontinuous, at these points
the derivative should be a Dirac pulse, but it isn't.
Dif numbers are printed with a trailing ~~ to
indicate that there is a "tail" of derivatives.
dVar function turns a number into a variable
number. This is the number with with respect to which
the derivaticve is computed.
deriv function is a simple utility to take the
derivative of a (single argument) function.
It is simply defined as
deriv f = val . df . f . dVar