Answer
\begin{aligned} f_{x}(x, y, z) &= y z \\ f_{y}(x, y, z) & =x z \\
f_{y y}(x, y, z) &= 0 \\ f_{x y}(x, y, z) & =z \\ f_{ y x}(x, y, z) &= z \\
f_{yy x}(x, y, z)&=0 \\ f_{x y y}(x, y, z)& =0 \\ f_{y xy}(x, y, z)& =0 \end{aligned}
And $$ f_{x y y} =f_{y x y}=f_{yy x}=0$$
Work Step by Step
Given $$f(x, y, z) =x y z$$
So, we have
\begin{aligned} f_{x}(x, y, z) &= \frac{\partial f(x,y)}{\partial x}=y z \\ f_{y}(x, y, z) &= \frac{\partial f(x,y)}{\partial y}=x z \\
f_{y y}(x, y, z) &= \frac{\partial^2 f(x,y)}{\partial y^2}=0 \\ f_{x y}(x, y, z) &= \frac{\partial^2 f(x,y)}{\partial y\partial x}=z \\ f_{ y x}(x, y, z) &= \frac{\partial^2 f(x,y)}{\partial z\partial y}=z \\
f_{yy x}(x, y, z)&= \frac{\partial^3 f(x,y)}{\partial x\partial y\partial y}=0 \\ f_{x y y}(x, y, z)&= \frac{\partial^3 f(x,y)}{\partial x\partial y\partial y}=0 \\ f_{y xy}(x, y, z)&= \frac{\partial^3 f(x,y)}{\partial y\partial x\partial y}=0 \end{aligned}
So, we get $$ f_{x y y} =f_{y x y}=f_{yy x}=0$$
