There are various papers on finding all the derivations of a nonassociative algebra and an anti-symmetrized algebra (see [2] [16]). We find all the derivations of the growing algebra W N (e ±x 1 x 2 x 3 , 0, 3) [1] with the set of all right annihilatorsis one and every derivation of the algebra W N (e ±x 1 x 2 x 3 , 0, 3) [1] is outer. We show that there is a class P of purely outer algebras in this work. PreliminariesLet N be the set of all non-negative integers and Z be the set of all integers. Let N + be the set of all positive integers. Let F be a field of characteristic zero and F • the set of all non-zero elements in F. Throughout the paper, we will assume that e is not the element of the field F. For n, t ∈ N, throughout the paper, m denotes a nonnegative integer such that m ≤ n + t.

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