Chapter 3.2, Problem 58ES

One model for the spread of information over time $t$ is given by

$\ln y-\ln \left(1-y\right)=\alpha t$

where $y$ is the fraction of the population with the information and $\alpha >0$ is a constant. Find an expression for $dy/dt$ and explain why $y$ is growing most rapidly at the moment when half the population has the information.