Alright, well currently you can use my prototype implementation from Leibniz.jl (which will not be needed in the future as the feature will be built into Grassmann.jl itself)
using Leibniz, DirectSum, SymEngine
import Leibniz: Monomial
printindices(V,D) = (io=IOBuffer();DirectSum.printindices(io,V,D,true);String(take!(io)))
Base.:*(d::Monomial{V,G,D,0} where G,r::Basic) where {V,D} = diff(r,symbols(Symbol(printindices(V,D))))
Base.:*(d::Monomial{V,1,D,1},r::Basic) where {V,D} = diff(r,symbols(Symbol((printindices(V,D)))))
x,y = symbols(:v1),symbols(:v2)
A = [x y^2; y^2 x*y]
b = [∂(ℝ^2,1),∂(ℝ^2,2)]
something like this should work with SymEngine
at least
julia> A*b
2-element Array{Any,1}:
1 + 2*v2
v1
is that what you wanted? as I said, Leibniz
is a prototype for what will be available in Grassmann