A new application of logarithmic differentiation is presented, which provides an alternative elegant proof of two basic rules of differentiation: the product rule and the quotient rule. The proof can intrigue students, help promote their critical thinking and rigorous reasoning and deepen their understanding of previously encountered concepts. The author also stresses the advantages of presenting the formulas for the product rule and the quotient rule in the forms recommended here over other forms which more commonly appear in the current mainstream calculus textbooks.