  We write the recently conjectured action for transformation of the ordinary
Born-Infeld action under the Seiberg-Witten map with one open Wilson contour in
a manifestly non-commutative gauge invariant form. This action contains the
non-constant closed string fields, higher order derivatives of the
non-commutative gauge fields through the $*_N$-product, and a Wilson operator.
We extend this non-commutative $D_9$-brane action to the action for $D_p$-brane
by transforming it under T-duality. Using this non-commutative $D_p$-brane
action we then evaluate the linear couplings of the graviton and dilaton to the
brane for arbitrary non-commutative parameters. By taking the Seiberg-Witten
limit we show that they reduce exactly to the known results of the
energy-momentum tensor of the non-commutative super Yang-Mills theory. We take
this as an evidence that the non-commutative action in the Seiberg-Witten limit
includes properly all derivative correction terms.
