Answer
$$\frac{\partial}{\partial x}f(x,y,z)=y^2(1-xz)e^{-xz};$$
$$\frac{\partial}{\partial y}f(x,y,z)=2xye^{-xz};$$
$$\frac{\partial}{\partial z}f(x,y,z)=-x^2y^2e^{-xz}.$$
Work Step by Step
Regard $y$ and $z$ as constant and differentiate with respect to $x$:
$$\frac{\partial}{\partial x}f(x,y,z)=\frac{\partial}{\partial x}(xy^2e^{-xz})=y^2\frac{\partial}{\partial x}(xe^{-xz})=y^2\big((x)'_xe^{-xz}+x(e^{-xz})'_x\big)=y^2(e^{-xz}+xe^{-xz}(-xz)'_x)=y^2(e^{-xz}+-xze^{-xz})=y^2(1-xz)e^{-xz}.$$
Regard $x$ and $z$ as constant and differentiate with respect to $y$:
$$\frac{\partial}{\partial y}f(x,y,z)=\frac{\partial}{\partial y}(xy^2e^{-xz})=xe^{-xz}\frac{\partial}{\partial y}y^2=2xye^{-xz}.$$
Regard $x$ and $y$ as constant and differentiate with respect to $z$:
$$\frac{\partial}{\partial z}f(x,y,z)=\frac{\partial}{\partial z}(xy^2e^{-xz})=xy^2\frac{\partial}{\partial z}e^{-xz}=xy^2e^{-xz}\frac{\partial}{\partial z}(-xz)=xy^2e^{-xz}(-x)=-x^2y^2e^{-xz}.$$
