Answer
$g_{rst}=e^{r}cos(st)-ste^{r}sin(st)$
Work Step by Step
Consider the function $g(r,s,t)=e^{r} sin(st)$
Let us start by finding $g_{r}(r,s,t)$ by differentiating $g(r,s,t) $ with respect to $r$ keeping $s$ and $t$ constant.
As we know
$g_{r}=\frac{∂}{∂r}g(r,s,t) $
$=\frac{∂}{∂r}[e^{r} sin(st)]$
$=e^{r} sin(st)$
Differentiate $g_{r}(r,s,t)$ with respect to $s$ keeping $r$ and $t$ constant .
$g_{rs}=\frac{∂}{∂s}[e^{r} sin(st)]=e^{r}tcos(st)$
Now,
$g_{rst}=\frac{∂}{∂t}[e^{r}tcos(st)]=-e^{r}stsin(st)+e^{r}cos(st)$
Hence, $g_{rst}=e^{r}cos(st)-ste^{r}sin(st)$