In this talk I will explore the subject of Bernoulli percolation on Galton-Watson trees.  Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results relating to the behavior of $g$ in the supercritical region.  These include an expression for the right derivative of $g$ at criticality in terms of the martingale limit of $T$, a proof that $g$ is infinitely continuously differentiable in the supercritical region, and a proof that $g'$ extends continuously to the boundary of the supercritical region.  Allowing for some mild moment constraints on the offspring distribution, each of these results is shown to hold for almost surely every Galton-Watson tree.  This is based on joint work with Marcus Michelen and Robin Pemantle.