In this paper we explicitly show the equivalence between the non-abelian Bo
rn-Infeld action, which was proposed by Tseytlin as an effective action on
several D-branes, and its noncommutative counterpart for slowly varying fie
lds. This confirms the equivalence between the two descriptions of the D-br
anes using an ordinary gauge theory with a constant B field background and
a noncommutative gauge theory, claimed by Seiberg and Witten. We also const
ruct the general forms of the 2n-derivative terms for non-abelian gauge fie
lds which are consistent with the equivalence in the approximation of negle
cting (2n + 2)-derivative terms.