Is it possible to get Mathematica to ''understand'' how differential operators work? i.e., to know that if Dx is the differential operator ''d/dx'', then Dx^2 is ''d^2/dx^2'', and, for example, Dx^2 (g*f), where g and f are both functions of x, is Dx(g'*f+f'*g) = g''*f+2g'*f'+g*f''
I've looked around on the online help, and can't seem to find out where to look for more information. Would I have to somehow define the d/dx operator in terms of its actions on functions, its commutation relations with other functions, and so on? How would I do that?
Any help is appriciated; if anyone's tried to do something like this with operators before, I'd love to hear from them.