This paper investigates the extension of Rademacher's theorem on the differentiability of Lipschitz functions to the setting of metric measure spaces that do not satisfy the doubling condition. The classical theorem and its initial generalizations by Pansu and Cheeger rely heavily on the geometric properties endowed by a doubling measure, which guarantees that the space is quantitatively similar to a Euclidean space at small scales. The absence of this condition presents significant analytical and geometric challenges. We explore a framework where the doubling condition is replaced by a weaker analytic assumption: the existence of a Poincaré inequality. This paper reviews the foundational work on analysis in metric spaces, introduces the necessary concepts such as upper gradients and Cheeger's differential structure, and outlines the construction of a differentiability space for Lipschitz functions. The main result is the formulation and proof of a Rademacher-type theorem in this non-doubling context, demonstrating that a Lipschitz function on a metric measure space supporting a Poincaré inequality is differentiable almost everywhere with respect to the underlying measure. We discuss the properties of the resulting differential and the implications of this extension for geometric measure theory and the study of partial differential equations in highly irregular settings.