Answer
\begin{aligned} dz=\left(8 x^{3} y-16 x y^{3}\right) d x+\left(2 x^{4}-24 x^{2} y^{2}\right) d y \end{aligned}
Work Step by Step
Given $$ z =2 x^{4} y-8 x^{2} y^{3}$$
Since $$dz=\frac{\partial z}{\partial x} dx+\frac{\partial z}{\partial y} dy,$$
$$\frac{\partial z}{\partial x} =8 x^{3} y-16 x y^{3}$$ and
$$\frac{\partial z}{\partial y} =2 x^{4} -24 x^{2} y^{2}$$
then we get
\begin{aligned} dz=\left(8 x^{3} y-16 x y^{3}\right) d x+\left(2 x^{4}-24 x^{2} y^{2}\right) d y \end{aligned}
