We will describe the derivations of a class of Lie superalgebras
generalizing the orthosymplectic Lie superalgebras. Examples are some
of the natural subalgebras of differential operators on the
(super)circle, recently studied by Kac-Wang-Yan in the classical case
and by Cheng-Wang in the supercase. Other examples are Lie algebras
graded by root systems of type B and D where our results give a
refinement of recent work of Benkart. In particular, we will give a
precise description of the derivations of those Lie algebras occurring
as the centreless core of extended affine Lie algebras.