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

For finding \(\displaystyle{f}_{{x}}\), we have to differentiate \(\displaystyle{f{{\left({x},{y}\right)}}}\) with respect to x as follows:

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

Now, for finding \(\displaystyle{f}_{{{x}{x}}}\), differentiate \(\displaystyle{f}_{{x}}\) with respect to \(\displaystyle{x}\).

Try in the same way for \(\displaystyle{f}_{{{y}{y}}}\).

For finding \(\displaystyle{f}_{{x}}\), we have to differentiate \(\displaystyle{f{{\left({x},{y}\right)}}}\) with respect to x as follows:

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

Now, for finding \(\displaystyle{f}_{{{x}{x}}}\), differentiate \(\displaystyle{f}_{{x}}\) with respect to \(\displaystyle{x}\).

Try in the same way for \(\displaystyle{f}_{{{y}{y}}}\).