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# How to define a derivative to be positive?

Posted 10 years ago
 Hi, I'm trying to define the derivative of a function H to be positive. I'm using H/:Positive[H'[x]]=True  But I get the error TagSetDelayed::tagpos: Tag H in Positive[H'[x]] is too deep for an assigned rule to be found.  I also tried defining a new function Hp which is H': Hp=H' Hp /: Positive[Hp[x]] := True  But then I get: TagSetDelayed::tagnf: Tag Hp not found in Positive[H'[x]].  I also tried wrapping Hp in Hold, but I get the "too deep" error as before. How can I declare it to be positive?
 Easy, type In[11]:= Positive[Derivative[1][h][x_]] ^:= True and check In[12]:= Positive[h'[p]] Out[12]= True In[13]:= NonNegative[h'[p]] Out[13]= NonNegative[Derivative[1][h][p]] In[15]:= Negative[h'[q]] Out[15]= Negative[Derivative[1][h][q]] In[16]:= Positive[k'[p]] Out[16]= Positive[Derivative[1][k][p]] In[19]:= Positive[h''[q]] Out[19]= Positive[(h^\[Prime]\[Prime])[q]] In[32]:= Positive[h'[q, p]] Out[32]= Positive[Derivative[1][h][q, p]]