Show that if $f \in C^{1}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime} \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbb{R}_{+} \mapsto P_{t}(f)(x)$ is of class $C^{1}$ on $\mathbb{R}_{+}$ and show that for all $t > 0$, we have $$\frac{\partial P_{t}(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x \mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1 - \mathrm{e}^{-2t}}} y\right) f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$
Show that if $f \in C^{1}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime} \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbb{R}_{+} \mapsto P_{t}(f)(x)$ is of class $C^{1}$ on $\mathbb{R}_{+}$ and show that for all $t > 0$, we have
$$\frac{\partial P_{t}(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x \mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1 - \mathrm{e}^{-2t}}} y\right) f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$