Find each derivative d) \( y=\frac{\ln x}{x+1} \)

To find the derivative of \( y = \frac{\ln x}{x+1} \), we’ll use the quotient rule, which states that if you have a function in the form of \( \frac{u}{v} \), the derivative is given by \( \frac{u'v - uv'}{v^2} \). Here, \( u = \ln x \) and \( v = x + 1 \). First, calculate the derivatives: - \( u' = \frac{1}{x} \) - \( v' = 1 \) Now, apply the quotient rule: \[ y' = \frac{\left(\frac{1}{x}(x + 1) - \ln x(1)\right)}{(x + 1)^2} \] This simplifies to: \[ y' = \frac{\frac{x + 1 - \ln x}{x}}{(x + 1)^2} = \frac{x + 1 - \ln x}{x(x + 1)^2} \] So, the derivative of \( y = \frac{\ln x}{x + 1} \) is: \[ y' = \frac{x + 1 - \ln x}{x(x + 1)^2} \] This is the final result for the derivative! Don't forget to check where this function is defined; it’s essential that \( x > 0 \) for \( \ln x \) to make sense.