A vertex coloring of a graph is called dynamic if the neighborhood of any vertex of degree at least 2 contains at least two vertices of distinct colors. Similarly to the chromatic number X(G) of a graph G, one can define its dynamic number X_d(G) (the minimum number of colors in a dynamic coloring) and dynamic chromatic number X₂(G) (the minimum number of colors in a proper dynamic coloring). We prove that X₂(G) ≤ X(G).X_d(G) and construct an infinite series of graphs for which this bound on X₂(G) is tight. For a graph G, set k=(G)We prove that X₂(G) ≤ (k+1)c. Moreover, in the case where k ≥ 3 and Δ(G) ≥ 3, we prove the stronger bound X₂(G) ≤ kc.