=\lim_{h\to 0+} \, \frac{2\cos \left(x^2+h x+\frac{h^2}{2}\right)}{\left(\sqrt{\sin (x+h)^2}+\sqrt{\sin \left(x^2\right)}\right)}\lim_{h\to 0+} \, \frac{\sin \left(\frac{h^2}{2}+h x\right)}{h}=\, \frac{2\cos \left(x^2\right)}{2\sqrt{\sin  x^2}}\lim_{h\to 0+} \, \frac{\sin \left(\frac{h^2}{2}+h x\right)}{h}=\frac{2\cos \left(x^2\right)}{2\sqrt{\sin  x^2}}\lim_{h\to 0+}  \frac{\frac{h^2}{2}+h x+o\left(\left(\frac{h^2}{2}+h x\right)^2\right)}{h}=\frac{\cos  x^2 }{\sqrt{\sin  x^2}}x=\frac{x \cos  x^2}{\sqrt{\sin  x^2}}