  Linear Logic refines Intuitionnistic Logic by taking into account the
resources used during the proof and program computation. In the past decades,
it has been extended to various frameworks. The most famous are indexed linear
logics which can describe the resource management or the complexity analysis of
a program. From an other perspective, Differential Linear Logic is an extension
which allows the linearization of proofs. In this article, we merge these two
directions by first defining a differential version of Graded linear logic:
this is made by indexing exponential connectives with a monoid of differential
operators. We prove that it is equivalent to a graded version of previously
defined extension of finitary differential linear logic. We give a denotational
model of our logic, based on distribution theory and linear partial
differential operators with constant coefficients.
