[tex]\it \left(\dfrac{x}{\sqrt{x^2+1}} \right)' = \dfrac{x'\sqrt{x^2+1}-x\cdot (\sqrt{x^2+1})'}{(\sqrt{x^2+1})^2} =
\\\;\\ \\\;\\
=\dfrac{\sqrt{x^2+1} - x \cdot\ \dfrac{2x}{ 2\sqrt{x^2+1}}}{x^2+1} =\dfrac{\sqrt{x^2+1} - \dfrac{x^2}{ \sqrt{x^2+1}}}{x^2+1} =
\\\;\\ \\\;\\
=\dfrac{x^2+1-x^2}{(x^2+1)\sqrt{x^2+1}} =\dfrac{1}{(x^2+1)\sqrt{x^2+1}} =\dfrac{1}{(x^2+1)^{\frac{3}{2}}}
[/tex]
