Defining an iterated differential operator with compositions

Question

I'm a bit new to Mathematica so this question might be a bit basic.
Still, I could not find an answer.
I wish to define a symbolic differential operator.
Currently, I do something like the following:
op[g_] = g'

Now, I would like this operator to work well with compositions.
For example I would like
op[g^2].

to return $2g'*g'$. However I only get $(g^2)'$.
Another thing I'm hoping will work is to compose the operator with itself, so that
op[op[g]]

will return $g''$.
I understand that the problem is that the argument 'g' was never defined to be a 'function of x'. But I'm sure how to define everything correctly.
Any help would be appreciated.

Ivan · Answer

You could use the function D:
op[g_] := D[g, x]

Then your result would be:
In[1]:= op[g[x]^2]
Out[2]= 2 g[x]g'[x]

And also
In[2]:= op[op[g[x]]]
Out[2]= g''[x]

Although I don't really see the point if the function D already exists.

Michael E2 · Answer

The problem is that g^2 does not represent the square of a function, that is, (g^2)[x] is not g[x]^2.  The system is not set up to operate that way. A similar thing could be said about 2 g * g', namely, that * does not perform the multiplication of functions.  To operate with functions, one can sometimes use Composition (@*) or pure functions.  Because of the way Derivative works, we are likely to end up with a pure-function expression anyway.
op[g_] := g'

To get g^2, compose Power[#, 2]& and g:
op[(#^2 &) @* g]

(*  2 g[#1] Derivative[1][g][#1] &  *)

Defining an iterated differential operator with compositions

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