Abstract: It is well known that, on a compact Riemannian manifold, minus t times the logarithm of the heat kernel converges uniformly to the energy function as t goes to zero. Malliavin and Stroock have used stochastic calculus to show that this limit commutes with spatial derivatives away from the cut locus, but one expects more complicated behavior at the cut locus. In this talk we will give formulas for the small time asymptotics of the gradient and the Hessian of the logarithm of the heat kernel which are valid everywhere on the manifold and which admit an appealing probabilistic interpretation. We will also show how these formulas can be used to study both the pointwise and the distributional limits of derivatives of the logarithm of the heat kernel.