We introduce a systematic approach to represent Leibniz’s nth-order diferential operator dn as the ratio of an infnite product of infnitesimal diference operators to an infnitesimal parameter. Because every diference operator can be expressed as a diference of two shift operators that translate the argument of a function by fnite amounts, Leibniz’s diferential operator dn is eventually expressed as the infnite product of infnitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This flls the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of diferential equations.