Answer
As a result $f_x=y^2e^{-xz}-xy^2ze^{-xz}$, $f_y=2xye^{-xz}$, $f_z=-x^2y^2e^{-xz}$.
Work Step by Step
$f(x,y,z)=xy^2e^{-xz}$
In order to find $f_x$ we treat $y$ and $z$ as constants and differentiate with respect to $x$.
$f_x=y^2e^{-xz}-xy^2ze^{-xz}$
In order to find $f_y$ and $f_z$ we do the same process.
$f_y=2xye^{-xz}$
$f_z=-x^2y^2e^{-xz}$