Let k, m be two positive integers with m ≤ k and let F be a family of zero-free meromorphic functions in a domain D, let h（z） ≡ 0 be a meromorphic function in D with all poles of h has multiplicity at most m. If, for each f ∈ F, f（k）（z） = h（z） has at most k- m distinct roots（ignoring multiplicity） in D, then F is normal in D. This extends the results due to Chang[1], Gu[3], Yang[11]and Deng[1]etc.