  We prove that the speed of a $\lambda$-biased random walk on a supercritical
Galton-Watson tree is differentiable for $\lambda$ such that the walk is
ballistic and obeys a central limit theorem, and give an expression of the
derivative using a certain $2$-dimensional Gaussian random variable. The proof
heavily uses the renewal structure of Galton-Watson trees that was introduced
by Lyons-Pemantle-Peres.
