Consider this multivariable function. f(x,y)=ye^(3x)+y^2 a) Find f_y(x,y) b) What is value of f_(xx)(0,3)?

Consider this multivariable function. f(x,y)=ye^(3x)+y^2 a) Find f_y(x,y) b) What is value of f_(xx)(0,3)?

Question
Multivariable functions
asked 2020-11-20
Consider this multivariable function. \(\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{{3}{x}}}+{y}^{{2}}\)
a) Find \(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}\)
b) What is value of \(\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}\)?

Answers (1)

2020-11-21
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) \(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={f}_{{x}}=\frac{\partial}{{\partial{x}}}{\left[{y}{e}^{{{3}{x}}}+{y}^{{2}}\right]}\)
\(\displaystyle={3}{e}^{{{3}{x}}}{y}+{0}\)
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={3}{y}{e}^{{{3}{x}}}\)
\(\displaystyle\frac{{\partial^{{2}}{f}}}{{\partial{x}^{{2}}}}={{f}_{{{x}{y}}}{\left({x},{y}\right)}}=\frac{\partial}{{\partial{x}}}{\left[{3}{y}{e}^{{{3}{x}}}\right]}\)
\(\displaystyle={9}{y}{e}^{{{3}{x}}}+{0}\)
\(\displaystyle{{f}_{{\times}}{\left({x},{y}\right)}}={9}{y}{e}^{{{3}{x}}}\)
\(\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}={9}\times{3}{e}^{{0}}={27}\)
0

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