Answer
$f_{xx}=0$
$f_{yy}=4xe^{-2y}$
$f_{xy}=f_{yx}=-2e^{-2y}$
Work Step by Step
Take the first partial derivatives of the given function. When taking partial derivative with respect to x, treat y as a constant, and vice versa:
$z_{x}=e^{-2y}$
$z_{y}=-2xe^{-2y}$
Then take the derivative of the first order partial derivatives to find second partial derivatives:
$f_{xx}=0$
$f_{yy}=4xe^{-2y}$
Second partial derivatives of first order partial derivative of x with respect to y and y with respect to x are the same:
$f_{xy}=f_{yx}=-2e^{-2y}$
