Re: Abs function

*To*: mathgroup at smc.vnet.net*Subject*: [mg47282] Re: Abs function*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Fri, 2 Apr 2004 03:30:30 -0500 (EST)*References*: <c4gf1i$276$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

nikmatz <nikmatz at math.ntua.gr> wrote: > i try to find the derivative of Abs[x] > and i have this > > In: D[Abs[x],x] > > Out: Abs`[x] > > i don't know the mean of ` > > please help The meaning of ` is, in this case, simply "derivative". But I'm sure that still leaves you unsatisfied. I'm glad you asked your question. It has now prompted me to write something which I had been intending to write for some time anyway: In this newsgroup we often get questions concerning differentiation or integration of user-defined piecewise-defined functions. The standard response seems to be to rewrite the function in terms of UnitStep. I was surprised and disappointed recently when I attempted to differentiate or integrate certain built-in functions, such as Abs and Sign. For example, I did not like In[1]:= Assuming[Element[x, Reals], D[Abs[x], x]] Out[1]= Derivative[1][Abs][x] since such an output is not helpful. One solution to this problem is the same as the solution when a piecewise-function is user-defined, namely, rewrite it in terms of UnitStep: In[2]:= RealAbs[x_]:= x*(2*UnitStep[x] - 1) In[2]:= Simplify[D[RealAbs[x],x]] Out[2]= -1 + 2*UnitStep[x] In[3]:= Simplify[Integrate[RealAbs[x],x]] Out[3]= (1/2)*x^2*(-1 + 2*UnitStep[x]) Of course, something similar can be done for other built-in functions such as Sign. Hope this helps, David Cantrell