Question Detail

To differentiate the given function, it's helpful to first use logarithm properties to simplify the expression:y = ln(x² / (x²+1))Using the property ln(a/b) = ln(a) - ln(b):y = ln(x²) - ln(x²+1)Using the property ln(a^p) = p ln(a):y = 2 ln(x) - ln(x²+1)Now, differentiate y with respect to x:dy/dx = d/dx (2 ln(x)) - d/dx (ln(x²+1))The derivative of ln(x) is 1/x, so d/dx (2 ln(x)) = 2/x.The derivative of ln(x²+1) uses the chain rule: d/dx (ln(f(x))) = f'(x) / f(x).Here, f(x) = x²+1, so f'(x) = 2x.Thus, d/dx (ln(x²+1)) = 2x / (x²+1).So, dy/dx = 2/x - 2x / (x²+1)To combine these terms, find a common denominator:dy/dx = (2(x²+1) - 2x(x)) / (x(x²+1))dy/dx = (2x² + 2 - 2x²) / (x(x²+1))dy/dx = 2 / (x(x²+1))Thus, option A is correct.