$f(x)={{e}^{\sqrt{2{{\sin }^{2}}\frac{x}{2}}}}={{e}^{\sqrt{2}\left| \sin \frac{x}{2} \right|}}\to x>0\to f(x)={{e}^{\sqrt{2}\sin \frac{x}{2}}}$  ${{{f}'}_{+}}(0)=\sqrt{2}\times \frac{1}{2}\times \cos \frac{x}{2}\times {{e}^{\sqrt{2}\sin \frac{x}{2}}}\left| \begin{matrix}    =\frac{\sqrt{2}}{2}\times 1\times 1=\frac{\sqrt{2}}{2}  \\    x=0  \\ \end{matrix} \right.$