Abstract
Summary–We bring to light a nearly forgotten identity found by Halphen in 1880. A special case is an expression for the nth derivative of . This formula was studied by Daboul, Mangaldan, Spivey, and Taylor in their recent paper in this Magazine. We reproduce Halphen’s original proof in a slightly improved form, which verifies his identity for monomial functions. More generally, we study derivatives of and establish a formula for the higher order derivatives of the product . As a consequence, we obtain a new proof of Halphen’s result which establishes the identity for arbitrary functions of sufficient smoothness.
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