For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. f(x,y)=x^2y-xy^2,where x = st and y = s/t

For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. f(x,y)=x^2y-xy^2,where x = st and y = s/t

Question
Derivatives
asked 2020-11-29
For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}{y}-{x}{y}^{{2}},{w}{h}{e}{r}{e}{x}={s}{t}{\quad\text{and}\quad}{y}=\frac{{s}}{{t}}\)

Answers (1)

2020-11-30
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>.
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