Step 1

\(\displaystyle{g{{\left({x},{y},{z}\right)}}}={2}{x}^{{{2}}}{y}-{3}{x}{z}^{{{4}}}+{10}{y}^{{{2}}}{z}^{{{2}}}\)

\(\displaystyle{g}_{{{x}}}={\left({2}\right)}{\left({2}\right)}{x}^{{{2}-{1}}}{y}-{3}{z}^{{{4}}}{x}^{{{1}-{1}}}+{0}\)

\(\displaystyle{g}_{{{x}}}={4}{x}{y}-{3}{z}^{{{4}}}\)

Step 2

\(\displaystyle{g}_{{{y}}}={2}{x}^{{{2}}}{y}^{{{1}-{1}}}-{0}+{10}{z}^{{{2}}}{\left({2}\right)}{y}^{{{2}-{1}}}\)

\(\displaystyle{g}_{{{y}}}{2}{x}^{{{2}}}+{20}{z}^{{{2}}}{y}\)

\(\displaystyle{g}_{{{y}}}{2}{x}^{{{2}}}+{20}{y}{z}^{{{2}}}\)

\(\displaystyle{g}_{{{z}}}={0}-{3}{x}{\left({4}\right)}{\left({z}\right)}^{{{4}-{1}}}+{10}{y}^{{{2}}}{\left({2}{z}^{{{2}-{1}}}\right)}\)

\(\displaystyle{g}_{{{z}}}=-{12}{x}{z}^{{{3}}}+{20}{y}^{{{2}}}{z}\)

\(\displaystyle{g{{\left({x},{y},{z}\right)}}}={2}{x}^{{{2}}}{y}-{3}{x}{z}^{{{4}}}+{10}{y}^{{{2}}}{z}^{{{2}}}\)

\(\displaystyle{g}_{{{x}}}={\left({2}\right)}{\left({2}\right)}{x}^{{{2}-{1}}}{y}-{3}{z}^{{{4}}}{x}^{{{1}-{1}}}+{0}\)

\(\displaystyle{g}_{{{x}}}={4}{x}{y}-{3}{z}^{{{4}}}\)

Step 2

\(\displaystyle{g}_{{{y}}}={2}{x}^{{{2}}}{y}^{{{1}-{1}}}-{0}+{10}{z}^{{{2}}}{\left({2}\right)}{y}^{{{2}-{1}}}\)

\(\displaystyle{g}_{{{y}}}{2}{x}^{{{2}}}+{20}{z}^{{{2}}}{y}\)

\(\displaystyle{g}_{{{y}}}{2}{x}^{{{2}}}+{20}{y}{z}^{{{2}}}\)

\(\displaystyle{g}_{{{z}}}={0}-{3}{x}{\left({4}\right)}{\left({z}\right)}^{{{4}-{1}}}+{10}{y}^{{{2}}}{\left({2}{z}^{{{2}-{1}}}\right)}\)

\(\displaystyle{g}_{{{z}}}=-{12}{x}{z}^{{{3}}}+{20}{y}^{{{2}}}{z}\)