Definition 5.1/5.2 is interesting. It defines derivative at point c, not the derivative function of f. Note that f1(x) is not equal to f'(x) for all x, but f1(c) = f'(c).
Yeah. The paper says: The derivative of f at c is written f'(c). The derivative of f, written f', is the function c -> f'(c).
So the derivative (f') is the result of substituting c for x in f1. For example, if f1 = (x -> x + c) then we would have f' = (c -> c + c) = (c -> 2c).
