  A symmetric tensor may be regarded as a partially symmetric tensor in several
different ways. These produce different notions of rank for the symmetric
tensor which are related by chains of inequalities. By exploiting algebraic
tools such as apolarity theory, we show how the study of the simultaneous
symmetric rank of partial derivatives of the homogeneous polynomial associated
to the symmetric tensor can be used to prove equalities among different
partially symmetric ranks. This approach aims to understand to what extent the
symmetries of a tensor affect its rank. We apply this to the special cases of
binary forms, ternary and quaternary cubics, monomials, and elementary
symmetric polynomials.
