Exact representation of the derivates of isotropic tensor functions with respect to the deformation gradient F

Expressions for derivatives of isotropic tensor functions with respect to the deformation tensor F are derived. Each derivative has the first representation in terms of eigenvectors; then, for computational conveniences, also a basis-free expression, in terms of eigenprojections, is reported. Further, in the same fashion, also the time derivatives are provided. In the paper, a short review of different approaches to the problem existing in literature is presented. In order to make the exposition self-contained, some backgrounds of tensor analysis are also given.