  In this paper, we analyze the existing rules for constructing derivatives of
the scalar and tensor functions of the tensor argument with respect to the
tensor argument and the theoretical positions underlying the construction of
these rules. We perform a comparative analysis of these rules and the results
obtained in the framework of these rules. Considering the existing approaches,
we pay due attention to the earliest of them which for some reason is not
reflected in later publications on the issue under consideration, and we give
to this approach the further development.We discuss the rules for
differentiation of tensor functions of the tensor argument and derivatives
obtained in the publications of other authors and examine them for compliance
with the construction rules and derivatives obtained in this article.
