We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by
$$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Calculate $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { \prime } ( x )$ and demonstrate that $\varphi$ is of class $C ^ { 1 }$ on $\mathbb { R }$.