Rewtire y

\(\displaystyle{y}={\left({\frac{{{x}^{{{2}}}+{2}}}{{{3}}}}\right)}^{{{2}}}={3}^{{-{2}}}{\left({x}^{{{2}}}+{2}\right)}^{{{2}}}\)

then

\(\displaystyle{y}'={\left[{3}^{{-{2}}}{\left({x}^{{{2}}}+{2}\right)}^{{{2}}}\right]}'\)

\(\displaystyle={3}^{{-{2}}}{\left({x}^{{{2}}}+{2}\right)}'\dot{{2}}{\left({x}^{{{2}}}+{2}\right)}\) Chain rule

\(\displaystyle={\frac{{{4}{x}{\left({x}^{{{2}}}+{2}\right)}}}{{{9}}}}\)

\(\displaystyle{y}={\left({\frac{{{x}^{{{2}}}+{2}}}{{{3}}}}\right)}^{{{2}}}={3}^{{-{2}}}{\left({x}^{{{2}}}+{2}\right)}^{{{2}}}\)

then

\(\displaystyle{y}'={\left[{3}^{{-{2}}}{\left({x}^{{{2}}}+{2}\right)}^{{{2}}}\right]}'\)

\(\displaystyle={3}^{{-{2}}}{\left({x}^{{{2}}}+{2}\right)}'\dot{{2}}{\left({x}^{{{2}}}+{2}\right)}\) Chain rule

\(\displaystyle={\frac{{{4}{x}{\left({x}^{{{2}}}+{2}\right)}}}{{{9}}}}\)