Question: Using partial derivatives as operators


I am trying to calculate the commutator $[v,w]:=vw-wv$.

For the sake of simplicity lets say

$$v=\dfrac{\partial}{\partial x}$$

$$w=x\dfrac{\partial}{\partial t}$$

I know how to calculate the commutator by hand, but I would like to learn how I can do these kind of calculations with maple.

**In this example the calculations by hand look like:**

$$[v,w]=\dfrac{\partial}{\partial x}\left( x\dfrac{\partial}{\partial t}\right)-x\dfrac{\partial}{\partial t}\dfrac{\partial}{\partial x}$$
$$=\dfrac{\partial}{\partial t}+x\dfrac{\partial^2}{\partial x \partial t}-x\dfrac{\partial^2}{\partial t \partial x}=\dfrac{\partial}{\partial t}$$

How can I "teach" maple to do this?

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