Let $\psi$ be the function defined on $\mathbb{R}$ by
$$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin ]-1,1[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$
Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.