Carvalho, Sonia2021-01-212021-01-212020-07-02CARVALHO, Sónia – An alternative proof on higher order derivatives of a multilinear map. Linear & Multilinear Algebra. ISSN 0308-1087. Vol. 68, N.º 7 (2020), pp. 1457-14640308-1087http://hdl.handle.net/10400.21/12665As a generalization of the formulas proved by Bhatia, Grover and Jain (Derivatives of tensor powers and their norms. Electron J Linear Algebra. 2013;26:604-619), in recent papers (The kth derivative of the immannant and the chi-symmetric tensor power of an operator. Electron J Linear Algebra. 2014;27:Article 18, On derivatives and norms of generalized matrix functions and respective symmetric powers. Electron J Linear Algebra. 2015;30:Article 22) Carvalho and Freitas obtained formulas for directional derivatives, of all orders, for generalized matrix functions and for every symmetric tensor power associated with a character xi of a subgroup G of the symmetric group S-m. Throughout our work, we used some well-known formulas for the derivatives of all orders of a multilinear map, since the maps that we studied are all multilinear. In this paper, we intend to present an alternative proof of these formulas, using the multilinearity argument.engMultilinear mapHigher order derivativesMultilinearity argumentjournal article10.1080/03081087.2018.15468161563-5139