A = {{1,2},{3,4}}
vec = {x^2, x^3}
D[vec.A.vec, x]
Or perhaps like this, again the table of derivatives:
xvec = {x1,x2}
Table[ D[xvec.A.xvec,x] ,{x,xvec}]
(all untested... one typo caught...)
aa[x_] = {{1, 2}, {3, 4}} x
bb[x_] = {x^2, x^3}
D[ a[x].b[x] , x, x] (* for any suitable tensors *)
% /. {a -> aa, b -> bb}
A = {{1,2},{3,4}}
vec = {x^2, x^3}
D[vec.A.vec, x]
Or perhaps like this, again the table of derivatives:
xvec = {x1,x2}
Table[ D[xvec.A.xvec,x] ,{x,xvec}]
(all untested... one typo caught...)