Step 1

The given function is \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{2}}}{\sin{{y}}}\).

This implies that,

\(\displaystyle{f}_{{{x}}}={2}{x}{\sin{{y}}}\)

\(\displaystyle{f}_{{{y}}}={x}^{{{2}}}{\cos{{y}}}\)

Step 2

Now obtain the second derivatives as follows.

\(\displaystyle{f}_{{\times}}={2}{\sin{{y}}}\)

\(\displaystyle{f}_{{{y}{y}}}=-{x}^{{{2}}}{\sin{{y}}}\)

\(\displaystyle{f}_{{{x}{y}}}={2}{x}{\cos{{y}}}\)

\(\displaystyle{f}_{{{y}{x}}}={2}{x}{\cos{{y}}}\)

The given function is \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{2}}}{\sin{{y}}}\).

This implies that,

\(\displaystyle{f}_{{{x}}}={2}{x}{\sin{{y}}}\)

\(\displaystyle{f}_{{{y}}}={x}^{{{2}}}{\cos{{y}}}\)

Step 2

Now obtain the second derivatives as follows.

\(\displaystyle{f}_{{\times}}={2}{\sin{{y}}}\)

\(\displaystyle{f}_{{{y}{y}}}=-{x}^{{{2}}}{\sin{{y}}}\)

\(\displaystyle{f}_{{{x}{y}}}={2}{x}{\cos{{y}}}\)

\(\displaystyle{f}_{{{y}{x}}}={2}{x}{\cos{{y}}}\)