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Re: bug? f'[x]'

• To: mathgroup at smc.vnet.net
• Subject: [mg90933] Re: bug? f'[x]'
• From: Ben <Benjamin.R.Lewis at gmail.com>
• Date: Wed, 30 Jul 2008 03:52:47 -0400 (EDT)
• References: <200807281153.HAA20595@smc.vnet.net> <344EC9C4-D0D2-4141-855E-DBE283AA7586@mimuw.edu.pl>

Mathematica does distinguish between f'[x] (evaluating the derivative)
and f[x]' (derivative of the evaluation), so it seems unreasonable
that this behaviour should change when f is written in terms of its
antiderivative (i.e., f->g').

In other words, what is the justification for Derivative[i]
[ (Derivative[j][g][x]) ][y] ever evaluating to Derivative[i+j][g][x]
[y]?

Even this behaviour is not consistent. Mathematica reverts to
mathematically correct behaviour again if g was explicitly specified.
For example, consider the function that takes an argument and returns
the operator for multiplication by the cube of that argument
(g=Function[x,Function[y,y*x^3]]). The derivative of its evaluation is
obviously (an operator that returns) a constant (namely the cube of
that argument, or zero for higher derivatives). By contrast, the
evaluation of the derivative is not a constant (it is the partial with
respect to the argument, an operator for multiplication by thrice the
squared argument; zero is not produced until the fourth derivative).
So, {g''[x][y],g'[x]'[y],g[x]''[y]}==={6xy,3x^2,0}.

Even Derivative[j]//Derivative[i] evaluates further, nonsensically
(garbage input ought only echo), as though a rule constraint had been
overlooked. I wondered if there could be scoping error, as internal
conversions involving #& are documented for Derivative. Also, this is
not the only case of odd behaviour involving the distinction between
numbers and operators:
http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/2b99d45979de6fd9/06396fcccf718c01

Ben

On Jul 29, 3:48 pm, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> Since, however, it is not documented
> (and illogical) it is certianly cannot be regarded as any kind of "bug".

• Follow-Ups:
  • Re: Re: bug? f'[x]'
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>

• References:
  • bug? f'[x]'
    • From: Benjamin.R.Lewis@gmail.com