Step 1
Given function is \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}},{w}{h}{e}{r}{e}{x}={s}{t},{y}={s}{<}\)</span>.
The all relevant second derivatives are \(\displaystyle{f}_{{\times}},{f}_{{{y}{y}}},{f}_{{{x}{y}}},{f}_{{{y}{x}}}\).
Step 2
Find partial derivative with respect to x for the function \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}}\).
\(\displaystyle\Rightarrow{f}_{{x}}={2}{x}{y}-{y}^{{2}}\)
Find partial derivative with respect to y for the function \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}}\).
\(\displaystyle\Rightarrow{f}_{{y}}={x}^{{2}}-{2}{x}{y}\)
Differentiate function \(\displaystyle{f}_{{x}}={2}{x}{y}-{y}^{{2}}\) withe respect to x to get \(\displaystyle{f}_{{\times}}\).
\(\displaystyle\Rightarrow{f}_{{\times}}={2}{y}\)
Substitute \(\displaystyle{y}=\frac{{s}}{{t}}\in{f}_{{\times}}={2}{y}\).
\(\displaystyle\Rightarrow{f}_{{\times}}=\frac{{{2}{s}}}{{t}}\)
Differentiate function \(\displaystyle{f}_{{y}}={x}^{{2}}−{2}{x}{y}\) with respect to y to get \(\displaystyle{f}_{{{y}{y}}}\).
\(\displaystyle\Rightarrow{f}_{{{y}{y}}}=-{2}{x}\)
Substitute x=st in \(\displaystyle{f}_{{{y}{y}}}=−{2}{x}\).
\(\displaystyle\Rightarrow{f}_{{{y}{y}}}=-{2}{s}{t}\)
Differentiate function \(\displaystyle{f}_{{y}}={x}^{{2}}−{2}{x}{y}\) with respect to x to get \(\displaystyle{f}_{{{x}{y}}}\).
\(\displaystyle\Rightarrow{f}_{{{x}{y}}}={2}{x}-{2}{y}\)
Substitute \(\displaystyle{x}={s}{t},{y}={s}{<}\in{f}_{{{x}{y}}}={2}{x}−{2}{y}\)</span>.
\(\displaystyle\Rightarrow{f}_{{{x}{y}}}={2}{s}{t}-\frac{{{2}{s}}}{{t}}\)
Differentiate function \(\displaystyle{f}{x}={2}{x}{y}−{y}^{{2}}\) with respect to y to get \(\displaystyle{f}_{{{y}{x}}}\).
\(\displaystyle\Rightarrow{f}_{{{y}{x}}}={2}{x}-{2}{y}\)
Substitute \(\displaystyle{x}={s}{t},{y}={s}{<}\in{f}_{{{y}{x}}}={2}{x}−{2}{y}\)</span>
\(\displaystyle\Rightarrow{f}_{{{y}{x}}}={2}{s}{t}-\frac{{{2}{s}}}{{t}}\).
Step 3
Therefore, all relevant second derivatives of function \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}},{w}{h}{e}{r}{e}{x}={s}{t},{y}={s}{<}{a}{r}{e},{f}_{{\times}}=\frac{{{2}{s}}}{{t}},{f}_{{{y}{y}}}=-{2}{s}{t},{f}_{{{x}{y}}}=\frac{{{2}{s}}}{{t}},{f}_{{{y}{x}}}=\frac{{{2}{s}}}{{t}}-{2}{s}{t}\)</span>.
Given function is \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}},{w}{h}{e}{r}{e}{x}={s}{t},{y}={s}{<}\)</span>.
The all relevant second derivatives are \(\displaystyle{f}_{{\times}},{f}_{{{y}{y}}},{f}_{{{x}{y}}},{f}_{{{y}{x}}}\).
Step 2
Find partial derivative with respect to x for the function \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}}\).
\(\displaystyle\Rightarrow{f}_{{x}}={2}{x}{y}-{y}^{{2}}\)
Find partial derivative with respect to y for the function \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}}\).
\(\displaystyle\Rightarrow{f}_{{y}}={x}^{{2}}-{2}{x}{y}\)
Differentiate function \(\displaystyle{f}_{{x}}={2}{x}{y}-{y}^{{2}}\) withe respect to x to get \(\displaystyle{f}_{{\times}}\).
\(\displaystyle\Rightarrow{f}_{{\times}}={2}{y}\)
Substitute \(\displaystyle{y}=\frac{{s}}{{t}}\in{f}_{{\times}}={2}{y}\).
\(\displaystyle\Rightarrow{f}_{{\times}}=\frac{{{2}{s}}}{{t}}\)
Differentiate function \(\displaystyle{f}_{{y}}={x}^{{2}}−{2}{x}{y}\) with respect to y to get \(\displaystyle{f}_{{{y}{y}}}\).
\(\displaystyle\Rightarrow{f}_{{{y}{y}}}=-{2}{x}\)
Substitute x=st in \(\displaystyle{f}_{{{y}{y}}}=−{2}{x}\).
\(\displaystyle\Rightarrow{f}_{{{y}{y}}}=-{2}{s}{t}\)
Differentiate function \(\displaystyle{f}_{{y}}={x}^{{2}}−{2}{x}{y}\) with respect to x to get \(\displaystyle{f}_{{{x}{y}}}\).
\(\displaystyle\Rightarrow{f}_{{{x}{y}}}={2}{x}-{2}{y}\)
Substitute \(\displaystyle{x}={s}{t},{y}={s}{<}\in{f}_{{{x}{y}}}={2}{x}−{2}{y}\)</span>.
\(\displaystyle\Rightarrow{f}_{{{x}{y}}}={2}{s}{t}-\frac{{{2}{s}}}{{t}}\)
Differentiate function \(\displaystyle{f}{x}={2}{x}{y}−{y}^{{2}}\) with respect to y to get \(\displaystyle{f}_{{{y}{x}}}\).
\(\displaystyle\Rightarrow{f}_{{{y}{x}}}={2}{x}-{2}{y}\)
Substitute \(\displaystyle{x}={s}{t},{y}={s}{<}\in{f}_{{{y}{x}}}={2}{x}−{2}{y}\)</span>
\(\displaystyle\Rightarrow{f}_{{{y}{x}}}={2}{s}{t}-\frac{{{2}{s}}}{{t}}\).
Step 3
Therefore, all relevant second derivatives of function \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}},{w}{h}{e}{r}{e}{x}={s}{t},{y}={s}{<}{a}{r}{e},{f}_{{\times}}=\frac{{{2}{s}}}{{t}},{f}_{{{y}{y}}}=-{2}{s}{t},{f}_{{{x}{y}}}=\frac{{{2}{s}}}{{t}},{f}_{{{y}{x}}}=\frac{{{2}{s}}}{{t}}-{2}{s}{t}\)</span>.
