Answer
$S_{u}= tan^{-1}(v\sqrt w)$,
$S_{v}= \frac{u\sqrt w}{(1+v^{2}w)}$
and
$S_{w}= \frac{uv}{(1+v^{2}w)2\sqrt w}$
Work Step by Step
Given: $S(u,v,w)=uarctan(v\sqrt w)$
The given function can also be written as
$S(u,v,w)=u tan^{-1}(v\sqrt w)$
Need to find first partial derivatives $S_{u}$,$S_{v}$ and $S_{w}$
Differentiate the function with respect to $u$ keeping $v$ and $w$ constant.
$S_{u}= tan^{-1}(v\sqrt w)$
Differentiate the function with respect to $v$ keeping $u$ and $w$ constant.
$S_{v}=u\times \frac{1}{(1+v^{2}w)}\times \sqrt w$
$= \frac{u\sqrt w}{(1+v^{2}w)}$
Differentiate the function with respect to $w$ keeping $u$ and $v$ constant.
$S_{w}=u\times \frac{1}{(1+v^{2}w)}\times v \times \frac{1}{2\sqrt w}$
$= \frac{uv}{(1+v^{2}w)2\sqrt w}$
Hence, $S_{u}= tan^{-1}(v\sqrt w)$,
$S_{v}= \frac{u\sqrt w}{(1+v^{2}w)}$
and
$S_{w}= \frac{uv}{(1+v^{2}w)2\sqrt w}$
