Let $M$ be a compact, connected, Riemannian manifold of dimension $d$, let $\{P_t:\,t>0\}$ denote the Markov semigroups on $C(M)$ determined by ${1\over 2}\Delta $, and let $p_t(x,y)$ denote the kernel (with respect to the Riemannian volume measure) for the operator $P_t$. (The existence of this kernel as a positive, smooth function is well-known, see e.g. [D].) Bismut's celebrated formula, presented in [B], equates $\nabla \log (p_t(\,\cdot \,,y))$ with certain stochastic integrals (see (20) below.) Various derivations of this formula and its extensions can be found in [AM], [EL] and [N]. In this note, we give a quick derivation of Bismut's and related formulae by lifting considerations to the bundle of orthonormal frames, using Bochner's identity, and applying a little elementary stochastic analysis. Some consequences of these identities are then explored. In particular, after deriving a standard logarithmic Sobolev inequality, we present (see (26)) a sharp pointwise estimate on the logarithmic derivative of the heat kernel in terms of known estimates on the heat kernel itself.