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AFAICT, the divided_difference method belongs to the class PolynomialRing_dense_mod_p. You attempt to apply it to a lsit of tuples of integers, for which it is not defined. Therefore, you must somehow give the crucial information (on whar ring are you working ?).
sage: R.<t>=PolynomialRing(RR)
sage: R.divided_difference([(0,1), (1,4.5), (3,14.5), (5,28.5), (6,37)])
[1.00000000000000,
3.50000000000000,
0.500000000000000,
0.000000000000000,
0.000000000000000]
HTH,
