6.4. Using the definitions of the left and right derivatives of a function $f(x)$ at a point $x_{0}$, find them for

$$
f(x)=\sqrt{1-e^{-x^{2}}} \text { and } x_{0}=0 \text {. }
$$
Solution.

$$
\begin{aligned}
f_{ \pm}(x)= & \lim _{\Delta x \rightarrow \pm 0} \frac{\sqrt{1-e^{-(0+\Delta x)^{2}}}-\sqrt{1-e^{-0^{2}}}}{\Delta x} \\
& =\lim _{\Delta x \rightarrow \pm 0} \frac{|\Delta x|}{\Delta x} \sqrt{\frac{1-e^{-(\Delta x)^{2}}}{(\Delta x)^{2}}}= \\
& = \pm \lim _{\Delta x \rightarrow \pm 0} \sqrt{\frac{1-e^{-(\Delta x)^{2}}}{(\Delta x)^{2}}}= \pm 1
\end{aligned}
$$