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In this paper we present the exact value for the norm of directionalderivatives, of all orders, for symmetric tensor powers of operators on finitedimensional vector spaces. Using this result we obtain an upper bound for thenorm of all directional derivatives of immanants.  This work is inspired in results by R. Bhatia, J. Dias da Silva, P. Groverand T. Jain.

The norm of the $k$-th derivative of the $\chi$-symmetric power of an operator