[tex](\frac{f}{g})'=\frac{f'g-fg'}{g^2}\\\\(\frac{x}{\sqrt{x^2+2x+2}})'=\frac{x'\sqrt{x^2+2x+2}-x(\sqrt{x^2+2x+2})'}{(\sqrt{x^2+2x+2})^2}=\frac{\sqrt{x^2+2x+2}-x\frac{(x^2+2x+2)'}{2\sqrt{x^2+2x+2}}}{x^2+2x+2}=\\\\=\frac{\sqrt{x^2+2x+2}-x\frac{2x+2}{2\sqrt{x^2+2x+2}}}{x^2+2x+2}=\frac{\sqrt{x^2+2x+2}-\frac{x^2+x}{\sqrt{x^2+2x+2}}}{x^2+2x+2}=\frac{\frac{x^2+2x+2-x^2-x}{\sqrt{x^2+2x+2}}}{x^2+2x+2}=\\\\=\frac{x+2}{(x^2+2x+2)(x^2+2x+2)^{\frac{1}{2}}}=\frac{x+2}{(x^2+2x+2)^{\frac{3}{2}}}[/tex]
