Question Detail

First, use the properties of logarithms to simplify the expression:ln(A/B) = ln(A) - ln(B).So, ln(x2/(x2+1)) = ln(x2) - ln(x2+1).Using another property, ln(xn) = n ln(x):ln(x2/(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)].d/dx [2ln(x)] = 2 ∙ (1/x) = 2/x.d/dx [ln(x2+1)] = (d/dx [x2+1]) / (x2+1) = 2x / (x2+1).So, the derivative is 2/x - 2x/(x2+1).Combine these terms over 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.