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Re: Re: Correction to "Fundamental Theorem of Calculus

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100763] Re: [mg100747] Re: Correction to "Fundamental Theorem of Calculus
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Sat, 13 Jun 2009 06:03:26 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <h0sbtl$hdk$1@smc.vnet.net> <200906120946.FAA27926@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

Yes. However, with those definitions of f and g:

   f'[x]
Sin[x^2]

   g'[x]
0

I believe that's the discrepancy to which the original poster refers. 
And this originates from what you'll see if you use the FullForm, the 
FullForm of h'[x] being Derivative[1][h][x]:

   Derivative[1][f]
(FresnelS^\[Prime])[Sqrt[2/\[Pi]] #1]&

   Derivative[1][g]
0&

The latter arises from:

   g[x]
1/2 Sqrt[\[Pi]] Erf[x]

So I don't understand why the derivative of g is the constantly 0 
function. After all, Mathematica DOES know:

   Derivative[1][Erf]
(2 E^-#1^2)/Sqrt[\[Pi]]&

And that surely is not the zero function!

Simon wrote:
> Hi Len,
> 
> Running both 6.0.3 and 7.0.1, I don't seem to get that problem:
> 
> In[1]:= f[x_]:=Integrate[Sin[t^2],{t,0,x}]
> 
> In[2]:= D[f[x],x]
> Out[2]= Sin[x^2]
> 
> In[3]:= g[x_]:=Integrate[Exp[-t^2],{t,0,x}]
> 
> In[4]:= D[g[x],x]
> Out[4]= E^-x^2
> 
> Simon
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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