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Re: How to use "Apply" to do differentiation ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg114514] Re: How to use "Apply" to do differentiation ?
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Mon, 6 Dec 2010 06:13:16 -0500 (EST)
  • References: <idd7ts$nja$1@smc.vnet.net> <idhjal$9cc$1@smc.vnet.net>

Am 06.12.2010 03:55, schrieb WetBlanket:
> On Dec 4, 3:16 am, Mayasky<alix.zh... at gmail.com>  wrote:
>> Something simple yet unbelievable occurred when I use:
>>
>> Apply[D[#, x]&, x^5]
>>
>> The output is invariably 1 whether I use x^5 or x^100.
>> Also I suggest you to try "Trace" command to see
>> the weirdness -- the output is messy if pasted as
>> text here.
>>
>> Finally I have to take a detour and use:
>> Nest[D[#, x]&, x^5, 1]
>>
>> I have been using Mathematica for several years and
>> never found that. I myself is wordless, but can anyone
>> explain that?
>
> You need to place x^5 in parenthesis
> Apply[D[#, x]&,{ x^5}]

You should not Apply D, its the wrong way of thinking.

In mainstream mathematics, D needs a function to operate on, providing 
the linear approximation of a function in direction locally, pointed to 
by the vector x->x+dx.

Applying D means to map the d_x-operation on the first argument of the 
function and projecting the result in the first Slot.

Compare eg.

Apply[D[#, x] &, f[a[x, y], b[x, y]]]
	-> Derivative[1, 0][a][x, y]

Map[D[#, x] &, f[a[x, y], b[x, y]]]
    -> f[Derivative[1, 0][a][x, y], Derivative[1, 0][b][x, y]]

Map[D[#, x] &, f[a[x, y], b[x, y]]]//First
   -> Derivative[1, 0][a][x, y]

-- 

Roland Franzius


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