Re: Re: Re: Question about derivatives

*To*: mathgroup at smc.vnet.net*Subject*: [mg51471] Re: [mg51466] Re: [mg51397] Re: [mg51384] Question about derivatives*From*: DrBob <drbob at bigfoot.com>*Date*: Tue, 19 Oct 2004 02:55:43 -0400 (EDT)*References*: <200410180150.VAA18514@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

>> I suggest that the answer be easier to find. Look to the subject of your own query. Does it describe what you want to do or how it can be done? If not, how can a search engine zero in on it? The same applies to archived answers to your question. >> why this behaviour is not built into Mathematica The same reason a million other behaviors aren't built in. They're not well defined, not practical, not high enough on anyone's priority list, .... et cetera. Bobby On Sun, 17 Oct 2004 21:50:08 -0400 (EDT), Ben Whale <ben.whale at anu.edu.au> wrote: > Thanks very much for the prompt reply. > > Before I asked this question, I searched, the archives, the FAQ and the > Mathematica Help extensively. If the question has been asked so many times, > I suggest that the answer be easier to find. I would also be interested in > why this behaviour is not built into Mathematica (or why there is no package > with new functions that does the same thing), where can I find the answers > to these questions? > > Ben > > -----Original Message----- > From: Andrzej Kozlowski [mailto:akoz at mimuw.edu.pl] To: mathgroup at smc.vnet.net > To: mathgroup at smc.vnet.net > Subject: [mg51471] [mg51466] [mg51397] Re: [mg51384] Question about derivatives > > On 15 Oct 2004, at 15:48, Ben Whale wrote: > >> Hi, >> I want to treat a function abstractly, I know that it is a function >> of n >> variables. I wish to take its derivative after addition or >> multiplication >> with another function. However Derivative does not return what I >> expect. >> Derivative[0,1][f+g] gives (f+g)^(0,1) instead of f^(0,1)+g^(0,1) and >> Derivative[0,1][fg] gives (fg)^(0,1) instead of f^(0,1)g+g^(0,1)f. >> Now >> perhaps I'm going about my problem in the wrong way, if I am please >> tell me. >> >> I also have some other questions about Derivative, >> Derivative[0,1][Composition[f,g]] gives f^(0,1)(g)g^(0,1), the correct >> result, why does Derivative know the chain rule and not the product >> rule? >> >> Also, in order to get the correct behaviour for Derivative[0,1][f+g], >> I can >> use the Map function, that is Map[Derivative[0,1],f+g] returns f^(0,1) >> +g^(0,1). Which is fine except that if I have the function 1+f and >> apply >> the same command to it I get 0& + f^(0,1), instead of f^(0,1). >> Clearly >> this is because of the presence of the command Function[0], however I >> have >> been unable to find a function that fixes this problem. >> >> In essence I want to do vector operations of the ring of functions, >> but find >> that doing this in Mathematica is hard. How can I do this easily? >> Surely >> someone else has figured out a convenient way of do this. >> >> Thanks, >> Ben >> >> >> > This question has been asked and answered many times (including quite > recently). The answer I am posting I have also already posted several > times. > For reasons which I do not want to argue about again Mathematica does > not implement the algebra of functions. that is, if f and g are > functions, whether defined using patterns as in f[x_]:= or as pure > functions using Function, f+g, f*g etc. are not is not treated by > Mathematica as functions. If you wan to change this you can always do > so: > > Unprotect[Times, Plus, Power]; > (a_?NumericQ*f_.)[x__] := a f[x]; > (f_ + g_)[x__] := f[x] + g[x]; > (f_*g_)[x__] := f[x]*g[x]; > (f_^n_?NumericQ)[x__] := f[x]^n > Protect[Times, Plus, Power]; > > Now you will get: > > > Derivative[0, 1][f + g] > > > Derivative[0, 1][f][#1, #2] + > Derivative[0, 1][g][#1, #2] & > > > Derivative[0, 1][f*g] > > > g[#1, #2]*Derivative[0, 1][f][#1, > #2] + f[#1, #2]* > Derivative[0, 1][g][#1, #2] & > > > You have to be aware however that in general it is not a good idea to > modify basic functions such as Times, Plus, Power. They are used by > practically every other Mathematica function and you can never tell if > having additional rules or modifications like those above will not > produce unexpected and unintended behaviour. > > Andrzej Kozlowski > > Chiba, Japan > http://www.akikoz.net/~andrzej/ > http://www.mimuw.edu.pl/~akoz/ > > > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**References**:**Re: Re: Question about derivatives***From:*"Ben Whale" <ben.whale@anu.edu.au>