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*: owner-wri-mathgroup 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