Procedure to assign the irreducible representation of a derivative operator
The transformation of a derivative operator under a linear transformation $\hat R $ is given by
$$
\frac{\partial}{\partial x_i} \rightarrow \sum_j R_{ji} \frac{\partial}{\partial x_j} = \sum R^T_{ij}\frac{\partial}{\partial  x_j}
$$
where $R_{nm}$ are the matrix elements of the matrix representation of the transformation operator in regard to cartesian coordiantes. The matrix is defined as the coordinate transformation matrix that takes the initial coordiantes to the transformed coordinates.
$$
\hat R : x \rightarrow  Rx
$$
The character of an representation with the derivative operators as basis is therefore the same as the character of the corresponding coordinate representation, since the trace of a matrix and a matrix transpose are identical. The derivative operators form thus a basis for the same irreducible representation as their coordinates and belong to the same irreducible representation.