Deduce that for $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ 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 $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$