We study derivations of the algebra of differential forms along the tangent bundle projection τ : TM [rarr] M and of the module of vector-valued forms along τ. It is shown that a satisfactory classification and characterization of such derivations requires the extra availability of a connection on TM. By composing the first prolongation of forms and fields along τ with the section of T²M representing a given second-order vector field Г, one obtains a corresponding calculus of forms on TM associated to this particular Г, which completely explains and generalizes earlier work on this subject.