Deduce that for $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth, we have
$$\forall t \in \mathbf{R}_+^*, \forall x \in \mathbf{R}, \quad \frac{\partial P_t(f)(x)}{\partial t} = L\!\left(P_t(f)\right)(x),$$
where $L(f)(x) = f''(x) - xf'(x)$.