Do you need to derive the Euler's formula for planar graphs with any number of components from the Euler's formula for graphs with one component? This is easy to do by induction on the number of components. If that number is k + 1 for k > 0, break the graph into two subgraphs, with k components and 1 component. Express the number of vertices, edges and regions in the whole graph through the corresponding numbers for the two subgraphs.