The k-th derivatives of the immanant and the chi-symmetric power of an operator

In recent papers, R. Bhatia, T. Jain and P. Grover obtained formulas for directional derivatives, of all orders, of the determinant, the permanent, the $m$-th compound map and the $m$-th induced power map. In this paper we generalize these results for immanants and for other symmetric powers of a matrix.