Let f, g be two transcendental meromorphic functions in C, let P be a polynomial of uniqueness for meromorphic functions in C and let alpha be a small meromorphic function with respect to f and g. If f' P'(f) and g' P'(g) share alpha counting multiplicity, then we show that f = g provided that the multiplicity orders of zeros of P' satisfy certain inequalities. There is no additional condition on alpha. We consider the particular case of entire functions. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.