# Find all first-order partial derivatives of the function at any

Find all first-order partial derivatives of the function at any point of the domain.
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{z}^{{{3}}}}}{{{x}^{{{2}}}}}}+{4}{y}-{x}^{{{16}}}+{2021}$$

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Jeffrey Parrish
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
$$\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{0}+{4}{\left({1}\right)}-{0}$$
$$\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{4}$$
$$\displaystyle{f}_{{{z}}}={0}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}+{0}$$
$$\displaystyle{f}_{{{z}}}=-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}$$