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Step 1: Identify the function and the differentiation rule needed. The given function is f(x) = (x^2 - 1)/(x^2 + 1). This is a quotient of two functions, so we will use the quotient rule: If f(x) = (g(x))/(h(x)), then f'(x) = (g'(x)h(x) - g(x)h'(x))/((h(x))^2). Step 2: Define g(x) and h(x) and find their derivatives. Let g(x) = x^2 - 1. Then g'(x) = (d)/(dx)(x^2 - 1) = 2x. Let h(x) = x^2 + 1. Then h'(x) = (d)/(dx)(x^2 + 1) = 2x. Step 3: Apply the quotient rule. Substitute g(x), h(x), g'(x), and h'(x) into the quotient rule formula: f'(x) = ((2x)(x^2 + 1) - (x^2 - 1)(2x))/((x^2 + 1)^2) Step 4: Simplify the numerator. f'(x) = (2x(x^2 + 1 - (x^2 - 1)))/((x^2 + 1)^2) f'(x) = (2x(x^2 + 1 - x^2 + 1))/((x^2 + 1)^2) f'(x) = (2x(2))/((x^2 + 1)^2) f'(x) = (4x)/((x^2 + 1)^2) Final answer: (4x)/((x^2 + 1)^2)