Answer
\[=\frac{1}{2}\sqrt{\frac{g\left( x \right)}{f\left( x \right)}}\left( \frac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{\left[ g\left( x \right) \right]}^{2}}} \right)\]
Work Step by Step
\[\begin{align}
& \frac{d}{dx}\left[ \sqrt{\frac{f\left( x \right)}{g\left( x \right)}} \right] \\
& \text{Rewrite} \\
& \frac{d}{dx}{{\left( \frac{f\left( x \right)}{g\left( x \right)} \right)}^{1/2}} \\
& \text{Diffetentiate by using the chain rule} \\
& =\frac{1}{2}{{\left( \frac{f\left( x \right)}{g\left( x \right)} \right)}^{-1/2}}\frac{d}{dx}\left( \frac{f\left( x \right)}{g\left( x \right)} \right) \\
& \text{Use the quotient rule} \\
& =\frac{1}{2}{{\left( \frac{f\left( x \right)}{g\left( x \right)} \right)}^{-1/2}}\left( \frac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{\left[ g\left( x \right) \right]}^{2}}} \right) \\
& \text{Simplifying} \\
& =\frac{1}{2}{{\left( \frac{g\left( x \right)}{f\left( x \right)} \right)}^{1/2}}\left( \frac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{\left[ g\left( x \right) \right]}^{2}}} \right) \\
& =\frac{1}{2}\sqrt{\frac{g\left( x \right)}{f\left( x \right)}}\left( \frac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{\left[ g\left( x \right) \right]}^{2}}} \right) \\
\end{align}\]
