Existence, Uniqueness & Bifurcations · Challenge problem

Let $y(t; r)$ solve the logistic equation $y' = y(r - y)$ with $y(0) = 1$. For $r = 1$ the solution is the equilibrium $y \equiv 1$. How sensitive is the solution to the parameter? Find $\dfrac{\partial y}{\partial r}$ at $r = 1$ and $t = \ln 2$.