We admit that $S$ is of class $C^1$ on $\mathbf{R}_+^*$ and that
$$\forall t \in \mathbf{R}_+^*, \quad S'(t) = \int_{-\infty}^{+\infty} \frac{\partial P_t(f)(x)}{\partial t}\left(1 + \ln\!\left(P_t(f)(x)\right)\right)\varphi(x)\,\mathrm{d}x.$$
Show that
$$\forall t \in \mathbf{R}_+^*, \quad S'(t) = \int_{-\infty}^{+\infty} L\!\left(P_t(f)\right)(x)\left(1 + \ln\!\left(P_t(f)(x)\right)\right)\varphi(x)\,\mathrm{d}x.$$