Question: How to define left and right derivatives as binary operators?

Dear Maple users,

Is there a way to describe derivatives treated as binary operators in infix notation, in which the derivatives is applied either to the left or right operands. This is useful, for example, when defining generalizations of the Poisson bracket. For a pair of functions f and g, the left and right derivatives are respectively defined as

f \stackrel{\leftarrow }{\partial }_x g = \frac{\partial f}{\partial x} \cdot g
f \stackrel{\rightarrow }{\partial }_x g = f \cdot \frac{\partial g}{\partial x}.

For example, it may be useful to calculate such object:

f exp(left_derivative_x right_derivative_x) g

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