Answer
$$\frac{\partial^2z}{\partial x^2}=0, \frac{\partial^2z}{\partial x\partial y}=\frac{\partial^2z}{\partial y\partial x}=6y, \frac{\partial^2z}{\partial y^2}=6x$$
Work Step by Step
The partial derivative with respect to $x$ is:
$$\frac{\partial z}{\partial x}=\frac{\partial}{\partial x}(3xy^2)=3y^2$$
The partial derivative with respect to $y$ is:
$$\frac{\partial z}{\partial y}=\frac{\partial}{\partial y}(3xy^2)=3x\cdot2y=6xy$$
The second partial derivatives are:
$$\frac{\partial ^2z}{\partial x^2}=\frac{\partial}{\partial x}\Big(\frac{\partial z}{\partial x}\Big)=\frac{\partial}{\partial x}(3y^2)=0$$
$$\frac{\partial ^2}{\partial y^2}=\frac{\partial}{\partial y}\Big(\frac{\partial z}{\partial y}\Big)=\frac{\partial }{\partial y}(6xy)=6x$$
$$\frac{\partial^2z}{\partial y\partial x}=\frac{\partial}{\partial y}\Big(\frac{\partial z}{\partial x}\Big)=\frac{\partial}{\partial y}(3y^2)=6y$$
$$\frac{\partial ^2z}{\partial x\partial y}=\frac{\partial}{\partial x}\Big(\frac{\partial z}{\partial y}\Big)=\frac{\partial }{\partial x}(6xy)=6y$$
We see that the second mixed partial derivatives are equal.
