Answer
\begin{align}d w=2 z^{3} y \cos x \ d x+2 z^{3} \sin x \ d y+6 z^{2} y \sin x\ d z\end{align}
Work Step by Step
Given $$ w =2 z^{3} y \ \sin x$$
Since $$dw=\frac{\partial w}{\partial x} dx+\frac{\partial w}{\partial y} dy+\frac{\partial w}{\partial z} dz,$$
$$\frac{\partial w}{\partial x} =2 z^{3} y \ \cos x ,$$
$$\frac{\partial w}{\partial y} =2 z^{3} \sin x$$
and
$$\frac{\partial w}{\partial z} =6 z^{2} y \ \sin x $$
then we get
\begin{align}d w=2 z^{3} y \cos x \ d x+2 z^{3} \sin x \ d y+6 z^{2} y \sin x\ d z\end{align}
