Small clarification just in case: the reason why you got NaN everywhere while Andy's first example (df.div(df2)) works for the first line is div tries to match indexes (and columns). In Andy's example, index 0 is found in both dataframes, so the division is made, not index 1 so a line of NaN is added. This behavior should appear even more obvious if you run the following (only the 't' line is divided):
df_a = pd.DataFrame(np.random.rand(3,5), index= ['x', 'y', 't'])
df_b = pd.DataFrame(np.random.rand(2,5), index= ['z','t'])
df_a.div(df_b)
So in your case, the index of the only row of df2 was apparently not present in df1. "Luckily", the column headers are the same in both dataframes, so when you slice the first row, you get a series, the index of which is composed by the column headers of df2. This is what eventually allows the division to take place properly.
For a case with index and column matching:
df_a = pd.DataFrame(np.random.rand(3,5), index= ['x', 'y', 't'], columns = range(5))
df_b = pd.DataFrame(np.random.rand(2,5), index= ['z','t'], columns = [1,2,3,4,5])
df_a.div(df_b)