By admitting that the result of question 7 is valid for the functions $P_t(f)$ and $1 + \ln(P_t(f))$, show that
$$\forall t \in \mathbf{R}_+^*, \quad -S'(t) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} \frac{P_t(f')(x)^2}{P_t(f)(x)}\,\varphi(x)\,\mathrm{d}x.$$