Question Detail

First, simplify the expression using logarithm properties: ln(a/b) = ln(a) - ln(b).So, ln(x2/(x2+1)) = ln(x2) - ln(x2+1) = 2ln(x) - ln(x2+1).Now differentiate with respect to x:d/dx [2ln(x) - ln(x2+1)] = d/dx [2ln(x)] - d/dx [ln(x2+1)]= 2/x - (1/(x2+1)) * (2x) (using chain rule for the second term)= 2/x - 2x/(x2+1)To combine these terms, find a common denominator:= (2(x2+1) - 2x*x) / (x(x2+1))= (2x2 + 2 - 2x2) / (x(x2+1))= 2 / (x(x2+1)).This matches option A.