1. It has been shown by Valiron [2] and Whittaker [3] that the derivative of a meromorphic function of finite order is of the same order as the function itself. This result, as pointed out by Whittaker, is equivalent to the following. THEOREM. If f (z) and g(z) are two integral functions of orders Pi and P2 with pl > p2, then f '(z)g(z) -f (z)g'(z) is of order pi. The proofs given by Valiron and Whittaker depend on meromorphic function theory, but in this paper I shall give a proof of the above theorem which depends entirely on integral function theory. 2. A number of lemmas are required and no proof will be given for the first of these as it is already well known. LEMMA 1 [1, p. 102 ]. Except for an exceptional set of intervals within which the variation of log r is finite zf'(z) = Nf(z){1 + o(1)}, I z = r where If(z) I = M(r, f) and N= N(r, f). LEMMA 2. There is an infinite sequence { Xi } J such that if Xi < r <Xi', a = (pi-e/2) (pi-e)-1, then log N(r, f) 2 (P1 e) log r. From the result [1, p. 33] lim sup I log N(r, f) }/logr= pi