Answer
$\displaystyle \frac{\partial w}{\partial x}=ze^{xyz}\cdot yz=yz^{2}e^{xyz}$,
$\displaystyle \frac{\partial w}{\partial y}=xz^{2}e^{xyz}$,
$\displaystyle \frac{\partial w}{\partial z}=e^{xyz}(xyz+1)$
Work Step by Step
$w=ze^{xyz}.$
Treat y and z as constant to calculate $\displaystyle \frac{\partial w}{\partial x}$
$\displaystyle \frac{\partial w}{\partial x} \stackrel{\text{chain rule} }{=} ze^{xyz}\cdot(yz)=yz^{2}e^{xyz}$
Treat x and z as constant to calculate $\displaystyle \frac{\partial w}{\partial y}$
$\displaystyle \frac{\partial w}{\partial y}\stackrel{\text{chain rule} }{=} ze^{xyz}\cdot(xz) =xz^{2}e^{xyz}$
Treat x and y as constant to calculate $\displaystyle \frac{\partial w}{\partial z}$
$\displaystyle \frac{\partial w}{\partial z}\stackrel{\text{product, chain rule} }{=}ze^{xyz}\cdot(xy)+e^{xyz}\cdot(1)=e^{xyz}(xyz+1)$
