Step 1

Given,

\(\displaystyle{g{{\left({y}\right)}}}={\left({y}-{4}\right)}{\left({2}{y}+{y}^{{{2}}}\right)}\)

Step 2

\(\displaystyle{g}'{\left({y}\right)}={\left({2}{y}+{y}^{{{2}}}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({y}-{4}\right)}+{\left({y}-{4}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({2}{y}+{y}^{{{2}}}\right)}\)

\(\displaystyle={\left({2}{y}+{y}^{{{2}}}\right)}{\left({1}\right)}+{\left({y}-{4}\right)}{\left({2}+{2}{y}\right)}\)

\(\displaystyle={\left({2}{y}+{y}^{{{2}}}\right)}+{\left({y}-{4}\right)}{\left({2}+{2}{y}\right)}\)

Hence,

\(\displaystyle{g}'{\left({y}\right)}={\left({2}{y}+{y}^{{{2}}}\right)}+{\left({y}-{4}\right)}{\left({2}+{2}{y}\right)}\)

Given,

\(\displaystyle{g{{\left({y}\right)}}}={\left({y}-{4}\right)}{\left({2}{y}+{y}^{{{2}}}\right)}\)

Step 2

\(\displaystyle{g}'{\left({y}\right)}={\left({2}{y}+{y}^{{{2}}}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({y}-{4}\right)}+{\left({y}-{4}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({2}{y}+{y}^{{{2}}}\right)}\)

\(\displaystyle={\left({2}{y}+{y}^{{{2}}}\right)}{\left({1}\right)}+{\left({y}-{4}\right)}{\left({2}+{2}{y}\right)}\)

\(\displaystyle={\left({2}{y}+{y}^{{{2}}}\right)}+{\left({y}-{4}\right)}{\left({2}+{2}{y}\right)}\)

Hence,

\(\displaystyle{g}'{\left({y}\right)}={\left({2}{y}+{y}^{{{2}}}\right)}+{\left({y}-{4}\right)}{\left({2}+{2}{y}\right)}\)