Answer
$z_{xx}=0$
$z_{xy}=-2e^{-2y}$
$z_{yx}=-2e^{-2y}$
$z_{yy}=4xe^{-2y}$
Work Step by Step
Given: $z=xe^{-2y}$
Partial differentiate with respect to $x$ is $z_{x}=e^{-2y}$ ...(1)
Partial differentiate equation (1) with respect to $x$ is $z_{xx}=0$
Partial differentiate equation (1) with respect to $y$ is $z_{xy}=-2e^{-2y}$
Partial differentiate with respect to $y$ is $z_{y}=-2xe^{-2y}$ ...(1)
Partial differentiate equation (2) with respect to $x$ is $z_{yx}=-2e^{-2y}$
Partial differentiate equation (2) with respect to $y$ is$z_{yy}=4xe^{-2y}$
