Formulae for the Derivatives of Heat Semigroups

Formulae for the derivatives of solutions of diffusion equations are derived which clearly exhibit, and allow estimation of, the equations′ smoothing properties. These also give formulae for the logarithmic gradient of the corresponding heat kernels, extending and giving a very elementary proof of Bismut′s well known formula. Corresponding formulae are derived for solutions of heat equations for differential forms and their exterior derivatives.