## Calculus 8th Edition

$S_{u}= tan^{-1}(v\sqrt w)$, $S_{v}= \frac{u\sqrt w}{(1+v^{2}w)}$ and $S_{w}= \frac{uv}{(1+v^{2}w)2\sqrt w}$
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 $S_{w}= \frac{uv}{(1+v^{2}w)2\sqrt w}$