       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
The University of Western Australia      (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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