Ornstein-Uhlenbeck gradient commutation identity ($(P_{t}f{)}^{\prime }=e^{-t}{P}_t(f^{\prime })$)

Differentiating the Mehler formula for the Ornstein-Uhlenbeck semigroup proves the identity for continuously differentiable functions with bounded derivative. Density extends it to the Gaussian Sobolev form domain. Together with a weighted Cauchy-Schwarz inequality, it yields the Ornstein-Uhlenbeck entropy dissipation identity proof of the Gaussian logarithmic Sobolev inequality.