Complex meromorphic functions f′P′(f),g′P′(g) sharing a small function

Let f,g be two transcendental meromorphic functions in C, let P be a polynomial of uniqueness for meromorphic functions in C and let α be a small meromorphic function with respect to f and g. If f′P′(f) and g′P′(g) share α 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 α. We consider the particular case of entire functions.