Expansion of the one-loop effective action in covariant derivatives

With an approach based on the heat-kernel representation, we show how to construct the expansion of the one-loop effective action in powers of covariant derivatives $D_{\mu }$ whenever it can be expressed in terms of an operator determinant of the form det(-$D^2$+V), where V is some positive Hermitian matrix-valued function. We present general expressions for the contributions to the effective Lagrangian in two and four covariant derivatives for four Euclidean space-time dimensions.