a) \(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}=\frac{{\partial{f}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{y}{e}^{{{3}{x}}}+{y}^{{2}}\right]}\)

\(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}={8}{e}^{{{3}{x}}}+{2}{y}\)

b) \(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={f}_{{x}}=\frac{\partial}{{\partial{x}}}{\left[{y}{e}^{{{3}{x}}}+{y}^{{2}}\right]}\)

\(\displaystyle={3}{e}^{{{3}{x}}}{y}+{0}\)

\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={3}{y}{e}^{{{3}{x}}}\)

\(\displaystyle\frac{{\partial^{{2}}{f}}}{{\partial{x}^{{2}}}}={{f}_{{{x}{y}}}{\left({x},{y}\right)}}=\frac{\partial}{{\partial{x}}}{\left[{3}{y}{e}^{{{3}{x}}}\right]}\)

\(\displaystyle={9}{y}{e}^{{{3}{x}}}+{0}\)

\(\displaystyle{{f}_{{\times}}{\left({x},{y}\right)}}={9}{y}{e}^{{{3}{x}}}\)

\(\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}={9}\times{3}{e}^{{0}}={27}\)

\(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}={8}{e}^{{{3}{x}}}+{2}{y}\)

b) \(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={f}_{{x}}=\frac{\partial}{{\partial{x}}}{\left[{y}{e}^{{{3}{x}}}+{y}^{{2}}\right]}\)

\(\displaystyle={3}{e}^{{{3}{x}}}{y}+{0}\)

\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={3}{y}{e}^{{{3}{x}}}\)

\(\displaystyle\frac{{\partial^{{2}}{f}}}{{\partial{x}^{{2}}}}={{f}_{{{x}{y}}}{\left({x},{y}\right)}}=\frac{\partial}{{\partial{x}}}{\left[{3}{y}{e}^{{{3}{x}}}\right]}\)

\(\displaystyle={9}{y}{e}^{{{3}{x}}}+{0}\)

\(\displaystyle{{f}_{{\times}}{\left({x},{y}\right)}}={9}{y}{e}^{{{3}{x}}}\)

\(\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}={9}\times{3}{e}^{{0}}={27}\)