$Containers is a set of heads over which FieldDerivative should distribute, in the following sense: Let c be a member of $Containers. Then FieldDerivative[c[f, g, h][x], x, {mu}] -> c[FieldDerivative[f[x], x, {mu}], FieldDerivative[f[x], x, {mu}], FieldDerivative[f[x], x, {mu}]].
$Containers\{\}