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Re: Derivative

• To: mathgroup at smc.vnet.net
• Subject: [mg43724] Re: Derivative
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 2 Oct 2003 02:51:18 -0400 (EDT)
• Organization: The University of Western Australia
• References: <blcqiv$p7d$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <blcqiv$p7d$1 at smc.vnet.net>,
 "extrabyte" <extrabyte22 at libero.it> wrote:

> I have these functions (see http://mynotebook.supereva.it)
> 
> g[x_,y_]:=(Exp[x-y]+1)^-1;
> 
> theta[x_,y_]:=ArcCosh[x*y];
> 
> F[y_]:=NIntegrate[g[x,y]*Sinh[theta[x,y]]*(Cosh[theta[x,y]])^2,{x,y,+oo},Wor
> kingPrecision->15, AccuracyGoal->20, MinRecursion->3, MaxRecursion->10]
> 
> I must now calculate:
> 
> f[y_]:=Evaluate[D[F[y]],y]
> 
> but Mathematica running..........

First use the fundamental theorem of calculus: for an arbitrary function 
h[x,y], we have

   D[Integrate[h[x,y],{x,y,Infinity}],y] ==
     Integrate[Derivative[0, 1][h][x, y], {x, y, Infinity}] - h[y, y]

Now define h,

  h[x_, y_] = Simplify[g[x, y] Sinh[theta[x, y]] Cosh[theta[x, y]]^2];

and compute its derivative with respect to y.

  dhdy[x_, y_] = Simplify[D[h[x,y],y]];

Now we can define f:

  f[y_] := NIntegrate[dhdy[x, y], {x, y, Infinity}, 
   WorkingPrecision -> 15, AccuracyGoal -> 20, 
   MinRecursion -> 3, MaxRecursion -> 10] - h[y, y]

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
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