Recursively differentiable quasigroups have been defined in [1], in connection with recursive MDS-codes. If C(n; f) is a recursive MDS-code of length n, defined by a binary quasigroup (Q; f), with jQj = q, then n · rmax + 3 and rmax · q ¡2, where rmax is the maximum order of recursive differentiability of (Q; f). It is known that rmax = q ¡ 2 when q is a power of a prime number, while in a general case to find rmax for a given q is an open problem [1,2,3]. Let (Q; f) be a binary quasigroup and s ¸ 0. The operation f(s), defined recursively as follows: f(0)(x; y) = f(x; y), f(1)(x; y) = f(y; f(x; y)), f(s)(x; y) = f((f(s¡2)(x; y); f(s¡1)(x; y), 8s ¸ 2, is called the recursive derivative of order s of (Q; f). A quasigroup (Q; f) is called recursively differentiable of order r if (Q; f(s)) is a quasigroup for all s · r. It is known that there exist recursively 1-differentiable binary quasigroups of every order q 6= 2; 6 and, possibly, q 6= 14; 18; 26; 42. We consider an extension of finite quasigroups using pairwise disjoint transversals and give necessary conditions when the prolongation of a recursively differentiable quasigroup is recursively 1-differentiable. Also we give an algorithm for the construction of finite linear recursively differentiable binary quasigroups of higher order, based on the following statements. Let consider on the set of integers Z the operation f(x; y) = (n ¡ k)x + y, where n ¸ 3, 1 · k · n ¡ 1. Then: 1. There exist us; vs; bs; cs 2 Z, such that f(s)(x; y) = usx + vsy, vs = nbs + cs, 8s ¸ 1, where c1 = ¡k + 1, c2 = ¡2k + 1, ci = ¡kci¡2 + ci¡1, 8i ¸ 3. 2. (Zn; f) is a recursively r-differentiable quasigroup if ((n ¡ k)vs; n) = 1 = (cs; n), for every 1 · s · r; 3. (Zpt ; f), where p is a prime and t ¸ 1, is a recursively r-differentiable quasigroup if (cs; n) = 1, for every 1 · s · r.