We define a new notion of fiber-wise linear differential operator on the total space of a vector bundle $E$. Our main result is that fiber-wise linear differential operators on $E$ are equivalent to (polynomial) derivations of an appropriate line bundle over $E^ast$. We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.
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