Question: How to extend a linear map by derivation rule on a polynomial algebra?

Hello!

I would appreciate any help on how to do computations on Maple of the following problem. 

Say we have 3 generators x,y,z and I define a map on polynomials in three variables over rational numbers L: Q[x,y,z] -> Q[x,y,z] on the generators, for example L(x)=xy, L(y)=1, L(z)=z^2. 

Then I need to compute L(xy+yz) and it needs to compute it using linearity and derivation (Leibniz rule):

L(x)y+xL(y)+L(y)z+yL(z)=xy^2+x+z+yz^2. 

I have in mind a recursive algorithm but I don't know enough syntacsys to implement it. 

P.S.: I need to compute Poisson brackets on a polynomial algebra for which there is no special way of doing it, right? So I though it must be easier for each basis element to come up with a linear map and extend it via derivations to know its action.