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Re: derivation of a picewise defined function
 To: mathgroup at smc.vnet.net
 Subject: [mg46124] Re: derivation of a picewise defined function
 From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
 Date: Sat, 7 Feb 2004 04:02:05 0500 (EST)
 References: <bvvnam$j4h$1@smc.vnet.net>
 Sender: ownerwrimathgroup at wolfram.com
"M.L. Bondar" <mbondar at win.tue.nl> wrote:
> I have the following problem. I need to work with the first derivative
> of a picewise defined function.
>
> A simple example :
>
> h[x_] := x /; 1 <= x
> h[x_] := E^(x  1) /; x <= 1
>
> I define
> j[x_] := h'[x]
>
> For 1<= x j[x] computes the values, j[x]=1 for all 1<=x, but for
> x<=1 it returnes me j[x]=h'[x].
>
> I need also the values of j for x<=1. How can I do that with
> Mathematica?
I suggest that you rewrite your function using UnitStep as
In[1]:= h[x_] := E^(x  1) + UnitStep[x  1](x  E^(x  1))
Then differentiating and simplifying gives what I think you want
In[2]:= D[h[x], x]
Out[2]= E^(1 + x) + (E^(1 + x) + x)*DiracDelta[1 + x] +
(1  E^(1 + x))*UnitStep[1 + x]
In[3]:= Simplify[%]
Out[3]= E^(1 + x) + (1  E^(1 + x))*UnitStep[1 + x]
David
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