We show the linear drift of the Brownian motion on the universal cover of a closed connected Riemannian manifold is C k´2 differentiable along any C k curve in the manifold of C k metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is C 1 differentiable along any C 3 curve of C 3 metrics with negative sectional curvature. We formulate the first derivatives of the linear drift and entropy, respectively, and show they are critical at locally symmetric metrics.