Question Detail

Let ( y = \ln\left(\frac{x^2}{x^2+1}\right) ). Using logarithm properties, ( y = \ln(x^2) - \ln(x^2+1) = 2\ln(x) - \ln(x^2+1) ).Now, differentiate with respect to ( x ):( \frac{dy}{dx} = \frac{d}{dx}(2\ln(x)) - \frac{d}{dx}(\ln(x^2+1)) )( \frac{dy}{dx} = \frac{2}{x} - \frac{1}{x^2+1} \cdot \frac{d}{dx}(x^2+1) )( \frac{dy}{dx} = \frac{2}{x} - \frac{1}{x^2+1} \cdot (2x) )( \frac{dy}{dx} = \frac{2}{x} - \frac{2x}{x^2+1} )Combine the fractions:( \frac{dy}{dx} = \frac{2(x^2+1) - 2x(x)}{x(x^2+1)} )( \frac{dy}{dx} = \frac{2x^2+2 - 2x^2}{x(x^2+1)} )( \frac{dy}{dx} = \frac{2}{x(x^2+1)} )