# 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. $f\left(x,y\right)=y{e}^{3x}+{y}^{2}$
a) Find ${f}_{y}\left(x,y\right)$
b) What is value of ${f}_{×}\left(0,3\right)$?
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a) ${f}_{y}\left(x,y\right)=\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}\left[y{e}^{3x}+{y}^{2}\right]$
${f}_{y}\left(x,y\right)=8{e}^{3x}+2y$
b) $\frac{\partial f}{\partial x}={f}_{x}=\frac{\partial }{\partial x}\left[y{e}^{3x}+{y}^{2}\right]$
$=3{e}^{3x}y+0$
$\frac{\partial f}{\partial x}=3y{e}^{3x}$
$\frac{{\partial }^{2}f}{\partial {x}^{2}}={f}_{xy}\left(x,y\right)=\frac{\partial }{\partial x}\left[3y{e}^{3x}\right]$
$=9y{e}^{3x}+0$
${f}_{xx}\left(x,y\right)=9y{e}^{3x}$
${f}_{xx}\left(0,3\right)=9×3{e}^{0}=27$