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Consider the Maclaurin series for \(\ln(1+x)\):$$ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots $$By differentiating both sides of this equation, find the Maclaurin series for the function \(f(x) = \frac{1}{1+x}\).
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