I want to know if it is possible to create a variable that is an operator. (Please note I only started programming Mathematica today) For example, the partial operator $ \partial_x$ i.e.

partX = D[#,x]& 

Ideally I would want to use it in the following way.

partX*(x^2) 

Which would return 2x. This is mostly so I can store it and manipulate it and use it when I want to.

On a related note, I was wondering if I could do this with lists as well.

del = {D[#,x]&, D[#,y]&, D[#,z]&} del*(xyz) 

Which would ideally return

{yz,xz,xy} 

I was wondering how this would work because I was also hoping to create operators like $ \partial_x\partial_y$ by multiplying the variables for each individually i.e.

partXY = D[#,x]&*D[#,y]& 

or

partX = D[#,x]& partY = D[#,y]& partXY = partX*partY 

Please let me know if you can answer any or all of these questions. Thanks!