Abstract: Let $X$ be a real uniformly convex and uniformly smooth Banach space and $C$ a nonempty closed and convex subset of $X$. Let $\Pi_C{:}~X\to C$ denote the generalized metric projection operator introduced by Alber in [1]. In this paper, we define the Gâteaux directional differentiability of $\Pi_{C.}$ We investigate some properties of the Gâteaux directional differentiability of $\Pi_C$. In particular, if $C$ is a closed ball, or a closed and convex cone (including proper closed subspaces), or a closed and convex cylinder, then, we give the exact representations of the directional derivatives of $\Pi_{C.}$ By comparing the results in [12] and this paper, we see the significant difference between the directional derivatives of the generalized metric projection operator $\Pi_{C}$ and the Gâteaux directional derivatives of the standard metric projection operator $P_{C.}$

Key words: uniformly convex and uniformly smooth Banach space, the generalized metric projection, directional differentiability of the generalized metric projection, directional derivative of the generalized metric projection