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Consider a power series

R.<x> = PowerSeries(SR)
f = 1 + O(x^2)
f.derivative()

This gives us O(x^1) as we would expect. However, if we do

f = O(x^0)
f.derivative()

we get O(x^-1) instead of O(x^0) again. Is this a bug or am I missing something?

I am trying to obtain a series solution to a differential equation. My hope was to substitute a power series with unknown coefficients into the differential equation to obtain a recursion relation for the coefficients. However, it seems I cannot create a power series with symbolic coefficients. For example:

var('C')
R.<x> = PowerSeriesRing(CC)
4 + O(x) # fine
C + O(x) # error

In particular,

TypeError: unsupported operand parent(s) for '+': 'Symbolic Ring' and 'Power Series Ring in x over Complex Field with 53 bits of precision'

So, my question is, can I define a power series with symbolic coefficients?