Mathematics in physics education: scanning historical evolution of the differential to find a more appropriate model for teaching differential calculus in physics

Despite its frequent use, there is little understanding of the concept of differential among upper high school and undergraduate students of physics. As a first step to identify the origin of this situation and to revert it, we have done a historic and epistemological study aimed at clarifying the role and the meaning of the differential in physics and at improving curricular and teaching models in the sense of Gilbert et al. (Gilbert J.K., Boulter C., & Rutherford, M.: 1998a, International Journal of Science Education 20(1), 83–97, Gilbert J.K., Boulter C., & Rutherford, M.: 1998b, International Journal of Science Education 20(2),187–203). We describe the contributions of Leibniz and Cauchy and stress their shortcomings, which are overcome by the alternative definition proposed by the French mathematician Frechet, dating from early 20th century. As a result of this study, we answer to some fundamental questions related to a proper understanding of the differential in physics education(for 17–19 years old students)