This paper is concerned with differentiation of interval functions appearing in interval analysis. Two definitions of a derivative are given; the first one uses an isometric restricted imbedding of the quasilinear space of intervals on the real line R, and the other definition is independent ofthat imbedding. Properties of those two concepts are investigated. Interval analysis was initiated by R. E. Moore [6] and has become an important tool in numerical problems. Further basic contributions are those by N. Apostolatos and U. Kulisch [1], E. Hansen [3], F. Kruckeberg [4], K. Nickel [7], and others. In the present paper we shall define and consider differentiation of interval functions; by definition, an interval function is a mapping of IiR) into itself, where IiR) is the set of all compact intervals on the real line R. For these intervals we use the notations A=[al, a2], B=[bx, b2], etc., a_=[a, a], etc., and the familiar addition and multiplication A+B: = {a+b\aeA,bB Secondary 28A15, 58C20.