Question

# 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)?

Multivariable functions
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 $$\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}$$?

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