Could you please provide the series in question? For me to help you write a function, f(x), that represents the derivative of the series, I need to know the specific terms of the series.

 In order to provide you with a function for the derivative of a given series, you would need to supply the actual series you are referring to. A series is expressed as the sum of terms according to a specific rule, and these terms are typically a function of an index, usually denoted as n. For example, a series can be written as:

∑ a_n for n = 1 to ∞

Where a_n represents the terms of the series. To find a derivative of a series with respect to x, we would first need to see how the terms a_n depend on x.

Once you provide the specific terms of the series as a function of x, we can proceed to differentiate it term-by-term if the series satisfies the conditions of term-by-term differentiation.

For instance, if your series is a power series of the form:

∑ c_n * (x - a)^n for n = 0 to ∞

Then the derivative of this series term-by-term would be:

∑ (n * c_n * (x - a)^(n-1)) for n = 1 to ∞

Please provide the series in question, so I may help you find the derivative function f(x) for the series.