The problem is that enumerate, by default, returns the indexes and items of the array passed to it with the indexes starting at 0. The array passed to it is the slice P[1:] = [5, 0, 2] and so enumerate generates, in turn, the tuples (0, 5), (1, 0) and (2, 2). However, for our derivative we need the indexes into the original list, P, giving (1, 5), (2, 0) and (3, 2). There are two alternatives: pass the optional argument start=1 to enumerate or add 1 to the default index:

>>> P = [4, 5, 0, 2]
>>> dPdx = []
>>> for i, c in enumerate(P[1:], start=1):
...     dPdx.append(i*c)
>>> dPdx
[5, 0, 6]

>>> P = [4, 5, 0, 2]
>>> dPdx = []
>>> for i, c in enumerate(P[1:]):
...     dPdx.append((i+1)*c)
>>> dPdx
[5, 0, 6]