  We consider biased random walks on the infinite cluster of a conditional bond
percolation model on the infinite ladder graph. Axelsson-Fisk and H\"aggstr\"om
established for this model a phase transition for the asymptotic linear speed
$\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value
$\lambda_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $\lambda\in
(0,\lambda_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if
$\lambda>\lambda_{\mathrm{c}}$.
  We show that the speed $\overline{\mathrm{v}}$ is continuous in $\lambda$ on
the interval $(0,\lambda_{\mathrm{c}})$ and differentiable on
$(0,\lambda_{\mathrm{c}}/2)$. Moreover, we characterize the derivative as a
covariance. For the proof of the differentiability of $\overline{\mathrm{v}}$
on $(0,\lambda_{\mathrm{c}}/2)$, we require and prove a central limit theorem
for the biased random walk. Additionally, we prove that the central limit
theorem fails to hold for $\lambda \geq \lambda_{\mathrm{c}}/2$.
