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MathGroup Archive 2006

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Re: Defining a derivative that distributes for a function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64253] Re: Defining a derivative that distributes for a function
  • From: Andres Corrada-Emmanuel <acorrada at physics.umass.edu>
  • Date: Wed, 8 Feb 2006 03:53:43 -0500 (EST)
  • References: <ds9mqp$48$1@smc.vnet.net> <43E886C2.6020309@metrohm.ch>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

Suppose f = MyIntegral and the the expression I'm differentiating is of the form
MyIntegral[expr1] + MyIntegral[expr2]. The two expressions are not equivalent at
all. 

Differentiating MyIntegral individually, I get

D[MyIntegral[expr1],x] = MyIntegral[ D[expr1,x] ]

But D[ MyIntegral[expr1] + MyIntegral[expr2],x] = D[expr1,x]*MyIntegral'[expr1]
+ D[expr2,x]*MyIntegral'[expr2]. This is not formally correct since D[expr1,x]
contains variables that need to remain under MyIntegrate.

Furthermore,if I don't define the derivative upvalue for f I get the expression
2xf'[x^2] + 4x^3f'[x^4]. So why is Mathematica not distributing the derivate
upvalue that I defined? And how do I get it to distribute the derivative correctly?

Quoting dh <dh at metrohm.ch>:

> Hi Andreas,
> both expressions are equivalent. The first one is obtained if the chain 
> rule is used, the seconde one if your rule is used.
> You can also directly verify the equivalence by noting that the solution 
> to your definition is: f[x_]= f[1] x.
> 
> Daniel
> 
> Andres Corrada-Emmanuel wrote:
> > Hello,
> > 
> > I'm trying to define a derivative for a function that distributes:
> > 
> > D[f[expr_],x_] ^:= f[D[expr,x]]
> > 
> > This gives me:
> > 
> > D[f[x^2],x] = f[2x]
> > 
> > and
> > 
> > D[f[x^4],x] = f[4x^3]
> > 
> > But D[f[x^2] + f[x^4],x] = 2xf'[x^2] + 4x^3f'[x^4] instead of the
> > desired:
> > 
> > D[f[x^2] + f[x^4],x] = f[2x] + f[4x^3]. Why? And how do I get the desired
> behaviour.
> > 
> > --------------------------------------
> > Andres Corrada-Emmanuel
> > Lecturer in Physics
> > Physics Department
> > University of Massachusetts at Amherst
> > --------------------------------------
> > 
> 
> 


--------------------------------------
Andres Corrada-Emmanuel
Lecturer in Physics
Physics Department
University of Massachusetts at Amherst
and
Program Manager
3-D Computer Vision and
Image Understanding Division
IAVO Research and Scientific
--------------------------------------


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