Question

If $f\left(x\right)=\sqrt{x}g\left(x\right)$, where $g\left(4\right)=2$ and $g^{\prime }\left(4\right)=3$, find $f^{\prime }\left(4\right)$.

Answer 1

$f^{\prime }\left(x\right)=\sqrt{x}{g}^{\prime }\left(x\right)+\frac{g\left(x\right)}{2\surd x}$
$f^{\prime }\left(4\right)=\sqrt{4}{g}^{\prime }\left(4\right)+\frac{g\left(4\right)}{2\surd 4}=2\times 3+\frac{2}{2\times 2}=6.5$