Answer
$$\nabla r =\left\langle z e^{y w}, xw z e^{y w},x e^{y w}, xy z e^{y w}
\right\rangle$$
Work Step by Step
Given
$$ r(x, y, z, w)=x z e^{y w}$$
Since
\begin{align*}
\frac{\partial r}{\partial x}&=z e^{y w}\\
\frac{\partial r}{\partial y}&=xw z e^{y w}\\
\frac{\partial r}{\partial z}&= x e^{y w}\\
\frac{\partial r}{\partial w}&= xy z e^{y w}
\end{align*}
Then
\begin{align*}
\nabla r&=\left\langle\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial r}{\partial z}, \frac{\partial r}{\partial w}\right\rangle\\
&=\left\langle z e^{y w}, xw z e^{y w},x e^{y w}, xy z e^{y w}
\right\rangle
\end{align*}
