I can't follow what Stewart is doing in his book. I can easily follow his work but his conclusion doesn't make any sense to me.
"Show that ever member of the family of functions
$y = \frac{1+ce^t}{1 - ce^t}$ is a solution of the differential equation $y' = \frac{1}{2} (y^2 - 1)$
He starts by differentiating the right side of the first term and then just setting that equal to the $y'$ from the question.
I do not see how these are equal or what this means or what is going on at all really, and the book doens't feel the need to explain this anyways, so maybe it isn't important and maybe I just need to memorize that a solution is just the differential. But I don't see hwy this is important or how this helps anything.
For his final answer he gets $y' = \frac{2ce^t}{(1-ce^t)^2}$