•First differences using RotateRight and Partition

examplelist = Table[a _ i, {i, 0, 7}]

{a _ 0, a _ 1, a _ 2, a _ 3, a _ 4, a _ 5, a _ 6, a _ 7}

diffVersion1[data_List] := Rest[data - RotateRight[data]]

diffVersion1[examplelist]

{a _ 1 - a _ 0, a _ 2 - a _ 1, a _ 3 - a _ 2, a _ 4 - a _ 3, a _ 5 - a _ 4, a _ 6 - a _ 5, a _ 7 - a _ 6}

diffVersion2[data_List] := Map[(Last[#] - First[#]) &, Partition[data, 2, 1]]

diffVersion2[examplelist]

{a _ 1 - a _ 0, a _ 2 - a _ 1, a _ 3 - a _ 2, a _ 4 - a _ 3, a _ 5 - a _ 4, a _ 6 - a _ 5, a _ 7 - a _ 6}