# Write first and second partial derivativesf(x,y)=2xy+9x^2y^3+7e^(2y)+16a)f_xb)f_(xx)c)f_(xy)d)f_ye)f_(yy)

Write first and second partial derivatives
$f\left(x,y\right)=2xy+9{x}^{2}{y}^{3}+7{e}^{2y}+16$
a)${f}_{x}$
b)${f}_{xx}$
c)${f}_{xy}$
d)${f}_{y}$
e)${f}_{yy}$
f)${f}_{yx}$

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unett

a) $\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}\left[2xy+9{x}^{2}{y}^{3}+7{e}^{2y}+16\right]$
${f}_{x}=2y+18x{y}^{3}+0+0=2y+18x{y}^{3}$
$⇒{f}_{x}=2y+18x{y}^{3}$
b) ${f}_{xx}=\frac{\partial {f}_{x}}{\partial x}=\frac{\partial }{\partial x}\left[9y+18x{y}^{3}\right]$
$=0+18{y}^{3}=18{y}^{3}$
${f}_{xx}=18{y}^{3}$
с) ${f}_{xy}=\frac{\partial {f}_{x}}{\partial y}=\frac{\partial }{\partial y}\left[2y+18x{y}^{3}\right]$
${f}_{xy}=2+18x\cdot 3{y}^{2}$
${f}_{xy}=2+54x{6}^{2}$
d) ${f}_{y}=\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}\left[2xy+9{x}^{2}{y}^{3}+4{e}^{2y}+16\right]$
${f}_{y}=2x+9{x}^{2}\cdot 3{y}^{2}+7\cdot 2{e}^{2y}+0$
${f}_{y}=9x+27{x}^{2}y+14{e}^{2}y$
e) ${f}_{yy}=\frac{\partial }{\partial y}\left[2x+27{x}^{2}{y}^{2}+14{e}^{2y}$
$=0+27{x}^{2}{x}^{{y}^{2}}+14\cdot 2{e}^{2y}$
${f}_{yy}=54{x}^{2}\cdot 2y+28{e}^{2y}$
f) ${f}_{yx}=\frac{\partial {f}_{y}}{\partial x}=\frac{\partial }{\partial x}\left[2x+27{x}^{{y}^{2}}+14{e}^{2y}\right]$
${f}_{yy}=2=54x{y}^{2}$