Re: Derivative[1] applied to numeric constants

*To*: mathgroup at smc.vnet.net*Subject*: [mg66780] Re: [mg66760] Derivative[1] applied to numeric constants*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 30 May 2006 05:48:23 -0400 (EDT)*References*: <200605291006.GAA07577@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On 29 May 2006, at 19:06, Andrew Moylan wrote: > Hi, > > Do numeric constants have special behaviour under the Derivative[1] > function? > > The number e.g. 2 is not defined as the function that always > returns 2: > > In[1]:=2[x] > Out[1]=2[x] > > But Derivative[1][2] is defined: > > In[2]:=2' > Out[2]=0& > > Could anyone explain why this is? Is this behaviour documented in the > help system? > > Cheers, > > Andrew > I can't give a definitive answer but note that we also have this closely related behaviour,that is documented: In[1]:= SetAttributes[f,Constant] In[2]:= Derivative[1][f] Out[2]= 0& or In[3]:= D[f[#]&,#] Out[3]= 0& and In[4]:= Dt[f[#]&] Out[4]= 0& This is consistent with the documentation for Constant: Functions f[ ? ] are taken to have zero total derivative if f has attribute Constant. If a symbol f with attribute Constant is treated in this way, it seems reasonable that genuine constants like 2 also are, although I can't find any explicit mention of this (perhaps nobody has considered the possibility that anyone might ask ;-)). But note that (from the Help for Derivative): Whenever Derivative[n][f] is generated, Mathematica rewrites it as D[f [#]&, {#, n}]. So Derivative[1][2] is D[2[#]&,{#,1}] and presumably, for the same reason as f[#]& when f has attribute Constant, this is taken to be 0. As for the "deeper reasons" why this is so: at the moment I can't think of one. I might play some role in the mechanism of functional differentiation or it could simply be a side-effect of something that does. (Of course, the function that always returns 2 is 2& and it has the same derivative as the abnormal "function" 2[#]& (or simple 2): Derivative[1][2&] 0& Derivative[1][2] 0& Andrzej Kozlowski

**References**:**Derivative[1] applied to numeric constants***From:*Andrew Moylan <andrew.moylan@anu.edu.au>