For the past century, Quantum Field Theory (QFT) has achieved spectacular success in particle physics. However, its reliance on the 'Mathematical Derivative,' which assumes spacetime is a perfect continuum, has plagued it with the non-physical problem of infinity. This paper proposes the 'Physical Derivative' as a new axiom, reflecting the fundamental discreteness of our universe. The Physical Derivative establishes the Planck length (l_p) as a natural lower limit for differentiation, thereby preventing the emergence of infinity from the very outset of the theory. Consequently, 'Renormalization' is no longer a mathematical sleight of hand to hide infinities, but is reinterpreted as an inevitable physical principle of scale transformation that arises when describing a discrete reality with the language of a continuum. We demonstrate the powerful explanatory power and utility of this new paradigm by applying it to solve (1) the electron's self-energy problem and (2) the cosmological constant problem.