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MathGroup Archive 2004

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Re: Re: Forcing a Derivative

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50834] Re: [mg50819] Re: Forcing a Derivative
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Thu, 23 Sep 2004 05:27:15 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200409220412.AAA18753@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

I see no difficulty in:

   f[x_] := x^2 + 7

   f'[x]
2x + 7

   f'[y]
2y + 7

even if one also has a definition for x[y_] or y[x_].  When f has one 
argument -- and at least so far that's all that's at issue here -- f' is 
the derivative of f, that is, stands for f'[#]&, so whatever the 
argument given in something of the form f'[expr], the result would be 
the value of the derivative at that input expr.

   x[y_]

Bill Rowe wrote:

> On 9/21/04 at 3:49 AM, murray at math.umass.edu (Murray Eisenberg)
> wrote:
> 
> 
>>So really the questions is:  WHY doesn't -- or, perhaps, why
>>shouldn't -- Mathematica understand such things as (f g)'?  Or (f +
>>g)' ,etc.???
> 
> 
> I think there are several issues here. First, how should Mathematica interpret f'? Sure, it is the derivative of f. But with respect to what? Yes, if I saw f was defined as x^2 + 7, I would assume f' meant df/dx. But is it reasonable for Mathematica to make this assumption when it is intended to be very general?
> 
> Note
> 
> f[x_] = x^2 + 7
> x^2 + 7
> 
> Head[f]
> Symbol
> 
> Head[x]
> Symbol
> 
> The point being x and f have the same head and are not clearly distinct. So, there could have been a previous definition such as x[y_] = 2 y. If this were the case, what should Mathematica do? Compute df/dx ignoring the definition for x or compute df/dy?
> 
> 
> Also, realize the original poster did define f as I did above, that is something to operated on an expression. Consequently, f' is really undefined since no argument as been supplied. Contrast
> 
> f[x]
> x^2 + 7
> 
> with 
> 
> f[2*x]
> 4*x^2 + 7
> 
> Again, this raises the question of what Mathematica should return for f'. It seems to me the only logical choice is to do what Mathematica currently does, return f' as unevaluated.
> 
> Finally note had the original poster defined f as a pure function i.e.,
> 
> f = #^2+7&
> 
> then f' and f'' return pure functions that are the expected derivatives, i.e.,
> 
> f'
> 2*#1 & 
> 
> f''
> 2 & 
> 
> When f is defined as a pure function in this manner, Mathematica knows f has only one argument and the only reasonable interpretation of f' is the derivative of f with respect to that argument. This is quite different than writing f[x_]= and later not supplying the argument x.
> --
> To reply via email subtract one hundred and four
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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