The generic Möbius transformation of the complex open unit disk includes a binary operation in the disk, called the Möbius addition. The extension of the Möbius addition to the ball of any real inner product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincaré ball model of hyperbolic geometry. The Möbius gyrovector spaces that emerge provide the setting for the Poincaré ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. The theories of gyrogroups and gyrovector spaces provide a new avenue for investigation, leading to a new approach to hyperbolic geometry and to new physics. The aim of this article is to extend the differential operation in vector spaces to a differential operation in gyrovector spaces, and to study its application to geodesics.