For a locally Lipschitz real-valued function f on R(n) and x, u in R(n) our main result implies that if x* is in Clarke’s subdifferential partial derivative J(x) "coming from the direction u" (in Chaney’s sense) such that x*(u) equals the directional derivative f’(x; u), then Chaney’s second-order directional derivative f "(x; x*, u), when it exists, coincides with the value at x* of the conjugate function of the Ben-Tal-Zowe second-order directional derivative, provided that this value is finite.