Question Detail
Question
Obtain d/dx {ln(x^2 / (x^2+1))}
Correct Answer
Option A is the correct answer.
Detailed Explanation
First, use the logarithm property ln(a/b) = ln(a) - ln(b) to rewrite the function: y = ln(x2) - ln(x2+1) = 2ln(x) - ln(x2+1). Now differentiate with respect to x: dy/dx = d/dx [2ln(x)] - d/dx [ln(x2+1)]. Using the chain rule, d/dx [ln(f(x))] = f'(x)/f(x), we get dy/dx = 2(1/x) - (2x/(x2+1)). Combine the terms: dy/dx = (2(x2+1) - 2x2) / (x(x2+1)) = (2x2 + 2 - 2x2) / (x(x2+1)) = 2 / (x(x2+1)).
Hint
Simplify the logarithmic expression first using logarithm properties before differentiating.