Question Detail
Question
Obtain d/dx {ln (x² / (x²+1))}
Correct Answer
Option A is the correct answer.
Detailed Explanation
First, simplify the logarithmic 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 this expression with respect to x:d/dx [2ln|x| - ln(x²+1)]= 2(1/x) - (1/(x²+1)) * (2x) (using chain rule for ln(u))= 2/x - 2x/(x²+1)To combine these fractions, find a common denominator, which is x(x²+1):= [2(x²+1) - 2x²] / [x(x²+1)]= [2x² + 2 - 2x²] / [x(x²+1)]= 2 / (x(x²+1))
Hint
Simplify the logarithmic function using log properties (ln(a/b) = ln(a) - ln(b)) before differentiating.