Question Detail

Let y = ln(x²/(x²+1)). We can simplify this expression using logarithm properties: ln(a/b) = ln(a) - ln(b).So, y = ln(x²) - ln(x²+1) = 2ln(x) - ln(x²+1).Now, differentiate y with respect to x:dy/dx = d/dx(2ln(x)) - d/dx(ln(x²+1))Using the rule d/dx(ln(u)) = u'/u:d/dx(2ln(x)) = 2(1/x) = 2/xd/dx(ln(x²+1)) = (2x)/(x²+1)So, dy/dx = 2/x - (2x)/(x²+1).To combine these terms, find a common denominator:dy/dx = (2(x²+1) - 2x*x) / (x(x²+1))dy/dx = (2x² + 2 - 2x²) / (x(x²+1))dy/dx = 2 / (x(x²+1)).