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Question about derivatives

• To: mathgroup at smc.vnet.net
• Subject: [mg51384] Question about derivatives
• From: "Ben Whale" <ben.whale at anu.edu.au>
• Date: Fri, 15 Oct 2004 02:48:03 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

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