Final answer:
To find f'(x) and f''(x) of the function f(x) = (x² - 1) / (x² + 1), we use the quotient rule. The derivative f'(x) is (4x) / (x² + 1)² and the second derivative f''(x) is (8x⁵ - 16x³ + 8x) / (x² + 1)⁴.
Step-by-step explanation:
To find the derivative (f'(x)) of the function f(x) = (x² - 1) / (x² + 1), we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), then f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x)²). Applying the quotient rule to f(x), we get:
f'(x) = ((2x * (x² + 1) - (x² - 1) * 2x) / (x² + 1)²). Simplifying this expression gives us:
f'(x) = (4x) / (x² + 1)²
To find the second derivative (f''(x)), we can differentiate f'(x). Differentiating f'(x) gives us:
f''(x) = (4 * (x² + 1)² - 4x * 2 * (x² + 1) * 2x) / (x² + 1)⁴
Simplifying further, we get:
f''(x) = (8x⁵ - 16x³ + 8x) / (x² + 1)⁴