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Function Repository Resource:

 ProductD

Source Notebook

Evaluate the product derivative of a function

Contributed by: Jan Mangaldan

ResourceFunction["ProductD"][f,x]

gives the product derivative of the function f with respect to x.

ResourceFunction["ProductD"][f,{x,n}]

gives the nth multiple product derivative of the function f with respect to x.

Details

The product derivative is also known as the multiplicative derivative or geometric derivative.
The first product derivative ResourceFunction["ProductD"][f[x],x] is defined as .
The second product derivative ResourceFunction["ProductD"][f[x],{x,2}] is the product derivative of the first product derivative; higher-order product derivatives are defined analogously.
The order of derivative n should be a non-negative integer.

Examples

Basic Examples (2) 

Product derivative of xa:

In[1]:=
ResourceFunction["ProductD"][x^a, x]
Out[1]=

Third product derivative of ex:

In[2]:=
ResourceFunction["ProductD"][E^x, {x, 3}]
Out[2]=

Scope (2) 

Product derivative of a polynomial:

In[3]:=
ResourceFunction["ProductD"][x^2 - 3 x + 1, x]
Out[3]=

Product derivative of the product logarithm:

In[4]:=
ResourceFunction["ProductD"][ProductLog[x], x]
Out[4]=

Properties and Relations (4) 

The product derivative of a function is defined as a limit:

In[5]:=
f[x_] := Gamma[x]
In[6]:=
ResourceFunction["ProductD"][f[x], x] == Limit[(f[x + h]/f[x])^(1/h), h -> 0]
Out[6]=

Higher-order product derivatives can be defined recursively:

In[7]:=
f[x_] := Gamma[x]
In[8]:=
With[{n = 3}, ResourceFunction["ProductD"][f[x], {x, n}] == ResourceFunction["ProductD"][ ResourceFunction["ProductD"][f[x], {x, n - 1}], x]]
Out[8]=
In[9]:=
ResourceFunction["ProductD"][f[x], {x, 0}] == f[x]
Out[9]=

Product rule for the product derivative:

In[10]:=
ResourceFunction["ProductD"][f[x] g[x], x] == ResourceFunction["ProductD"][f[x], x] ResourceFunction["ProductD"][ g[x], x] // Simplify
Out[10]=

Quotient rule for the product derivative:

In[11]:=
ResourceFunction["ProductD"][f[x]/g[x], x] == ResourceFunction["ProductD"][f[x], x]/ ResourceFunction["ProductD"][g[x], x] // Simplify
Out[11]=

Neat Examples (1) 

A function whose product derivative is itself:

In[12]:=
ResourceFunction["ProductD"][E^E^x, x]
Out[12]=

Version History

  • 1.0.0 – 02 March 2021

Source Metadata

  • Citation:
    • Grossman M., Katz R., Non-Newtonian Calculus. Lee Press, Pigeon Cove, Massachusetts, 1972.

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