How do I define the $n$th product derivative of a function in Mathematica?
The first product derivative $f^\ast$ of a function $f$ is $$ f^\ast(x)=\exp\left(\frac{f^\prime(x)}{f(x)}\right) $$ The $n$th product derivative is the result of applying this operator $n$ times.
This is my attempt at a recursive definition:
In[111]:= Clear[prodd]
prodd[f_, n_] := prodd[e^(f'/f), n - 1]; prodd[f_, 0] = f
Out[112]= f
In[113]:= prodd[E^x, 1](* Should print E *)
Out[113]= e^(E^-x Derivative[1][(E^x)])
In[114]:= prodd[E^E^x, 1](* Should print E^E^x *)
Out[114]= e^(E^-E^x Derivative[1][(E^E^x)])
In[114]:= prodd[E^x, 2](* Should print 1 *)