Network coding is a novel method for transmitting data, which has been recently proposed, and has been shown to have potential to improve wireless network performance. In this article, we study using network coding for one specific case of multicast, broadcasting. Precisely, we focus on (energy-)efficient broadcasting in a multi-hop wireless networks: transmitting data from one source to all nodes with a small number of retransmissions. It is known that finding an efficient method to broadcast, is essentially summarized in selecting proper transmission rates of each node. Our contribution, is proposing a simple and efficient method for determining a rate selection. Our method adapts dynamically and uses only local dynamic information of neighbors: Dynamic Rate Adaptation from Gap with Other Nodes (D.R.A.G.O.N.). The rationale of this rate selection method is detailed from some logical arguments. Experimental results illustrate the behavior of the method, and its excellent performance.