Re: RE:? D[f,{x,n}]

*To*: mathgroup at smc.vnet.net*Subject*: [mg25570] Re: [mg25559] RE:[mg25495] ? D[f,{x,n}]*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sun, 8 Oct 2000 00:41:57 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

on 10/7/00 4:36 PM, Ersek, Ted R at ErsekTR at navair.navy.mil wrote: > seems to prove that D[f, {x, n}] evaluates > Nest[ D[#, x]&, f, n ] > or something equivalent. > ----------------------------- I think the point is (as already has been pointed out by Carl Woll) that: In[22]:= MatchQ[Unevaluated[D[f, {x, n}]], Unevaluated[D[f_, y_]]] Out[22]= True We can see what happens in your case simply by looking at Trace: In[28]:= Trace[D[(x + 2)^4, {x, 2}]] Out[28]= {{{HoldForm[x + 2], HoldForm[2 + x]}, HoldForm[(2 + x)^4]}, HoldForm[D[(2 + x)^4, {x, 2}]], {HoldForm[flag], HoldForm[True]}, HoldForm[Block[{flag}, D[Expand[(2 + x)^4], {x, 2}]]], {{HoldForm[Expand[(2 + x)^4]], HoldForm[16 + 32*x + 24*x^2 + 8*x^3 + x^4]}, HoldForm[D[16 + 32*x + 24*x^2 + 8*x^3 + x^4, {x, 2}]], HoldForm[48 + 48*x + 12*x^2]}, HoldForm[48 + 48*x + 12*x^2]} As you can see, D[(2 + x) , {x, 2}] was found to match the rule for D[f_,x_]/;flag and the rule was simply applied. There was need to use Nest, and so on. This is just a case of pattern matching doing its job. A similar approach can be used to show that when Mathematica evaluates Derivative[n][f] it does not do actually do this by nesting Derivative[1]'s. Here is a rule for Derivative that is, essentially, equivalent to the above one for D (note that there is no need to use Unprotect as Derivative does not have the Protected Attribute!): In[1]:= Derivative[1][Function[p_]] /; flag := Block[{flag}, Derivative[1][Function @@ {Expand[p] }]] In[2]:= flag = True; This works just as above: In[3]:= (# + 2)^3 &'[x] Out[3]= 2 12 + 12 x + 3 x But: In[4]:= ((# + 2)^3 &)''[x] Out[4]= 6 (2 + x) While on the other hand In[5]:= ((# + 2)^3 &')'[x] Out[5]= 12 + 6 x Of course what one should do is use a more general pattern: In[6]:= Derivative[n_][Function[p_]] /; flag := Block[{flag}, Derivative[n][Function @@ {Expand[p] }]] so that: In[7]:= ((# + 2)^5 &)'''[x] Out[7]= 2 240 + 240 x + 60 x -- Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/ http://sigma.tuins.ac.jp/