Step 1

To find all first - order partial derivatives of the function :-

\(\displaystyle{f{{\left({x},{y},{z}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{z}^{{{3}}}}}{{{x}^{{{2}}}}}}+{4}{y}-{x}^{{{16}}}+{2021}\)

\(\displaystyle{f}_{{{x}}}={12}{x}^{{{2}}}{y}^{{{2}}}-{0}-{\frac{{{2}{z}^{{{3}}}}}{{{x}^{{{3}}}}}}+{0}-{16}{x}^{{{15}}}+{0}\)

\(\displaystyle{f}_{{{x}}}={12}{x}^{{{2}}}{y}^{{{2}}}-{\frac{{{2}{z}^{{{3}}}}}{{{x}^{{{3}}}}}}-{16}{x}^{{{15}}}\)

Step 2

\(\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{0}+{4}{\left({1}\right)}-{0}\)

\(\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{4}\)

\(\displaystyle{f}_{{{z}}}={0}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}+{0}\)

\(\displaystyle{f}_{{{z}}}=-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}\)

To find all first - order partial derivatives of the function :-

\(\displaystyle{f{{\left({x},{y},{z}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{z}^{{{3}}}}}{{{x}^{{{2}}}}}}+{4}{y}-{x}^{{{16}}}+{2021}\)

\(\displaystyle{f}_{{{x}}}={12}{x}^{{{2}}}{y}^{{{2}}}-{0}-{\frac{{{2}{z}^{{{3}}}}}{{{x}^{{{3}}}}}}+{0}-{16}{x}^{{{15}}}+{0}\)

\(\displaystyle{f}_{{{x}}}={12}{x}^{{{2}}}{y}^{{{2}}}-{\frac{{{2}{z}^{{{3}}}}}{{{x}^{{{3}}}}}}-{16}{x}^{{{15}}}\)

Step 2

\(\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{0}+{4}{\left({1}\right)}-{0}\)

\(\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{4}\)

\(\displaystyle{f}_{{{z}}}={0}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}+{0}\)

\(\displaystyle{f}_{{{z}}}=-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}\)