A long-standing scientific problem about ΔG and ∂G/∂ξ, involving the Gibbs-Duhem equation, demonstrates that wrong concepts are sometimes dominant in the literature even though corrections are readily available. This fact shows that the Gibbs-Duhem equation and especially its applications are among the most difficult topics in thermodynamics. Moreover, it demonstrates that there are often weak points in existing methods which necessitate new perspectives. There exist several different derivations of the Gibbs-Duhem equation. A novel mathematical derivation of the equation is developed here and is demonstrated to be more natural, direct, and intuitive than its thermodynamics counterpart. Different perspectives of mathematical and thermodynamic derivations are illustrated along with discussions concerning methods for solving specific scientific problems. The introduction of different self-consistent methods to solve a specific problem is vital in providing researchers with a variety of perspectives, thus, Legendre transformations are used here to introduce thermodynamic functions since the method is unfamiliar to many researchers in this field, also because it is a good method