Answer
\[\frac{{{g}^{2}}\left( x \right)f''\left( x \right)-g\left( x \right)f\left( x \right)g''\left( x \right)-2g'\left( x \right)g\left( x \right)f'\left( x \right)+2f\left( x \right){{\left( g'\left( x \right) \right)}^{2}}}{{{\left[ g\left( x \right) \right]}^{3}}}\]
Work Step by Step
\[\begin{align}
& \frac{d}{dx}\left[ \frac{f\left( x \right)}{g\left( x \right)} \right] \\
& \text{By the quotient rule, we know that} \\
& \frac{d}{dx}\left[ \frac{f\left( x \right)}{g\left( x \right)} \right]=\frac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{\left[ g\left( x \right) \right]}^{2}}} \\
& \text{Calculate the second derivative by the quotient rule} \\
& =\frac{{{\left[ g\left( x \right) \right]}^{2}}\frac{d}{dx}\left[ g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{4}}} \\
& -\frac{\left[ g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right) \right]\frac{d}{dx}{{\left[ g\left( x \right) \right]}^{2}}}{{{\left[ g\left( x \right) \right]}^{4}}} \\
& \text{Differentiate} \\
& =\frac{{{\left[ g\left( x \right) \right]}^{2}}\left[ g\left( x \right)f''\left( x \right)+f'\left( x \right)g'\left( x \right)-f\left( x \right)g''\left( x \right)-g'\left( x \right)f'\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{4}}} \\
& -\frac{\left[ g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right) \right]\left[ 2g\left( x \right)g'\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{4}}} \\
& \text{Simplifying} \\
& =\frac{\left[ g\left( x \right)f''\left( x \right)-f\left( x \right)g''\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{2}}} \\
& -\frac{\left[ g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right) \right]\left[ 2g'\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{3}}} \\
& =\frac{{{g}^{2}}\left( x \right)f''\left( x \right)-g\left( x \right)f\left( x \right)g''\left( x \right)-2g'\left( x \right)g\left( x \right)f'\left( x \right)+2f\left( x \right){{\left( g'\left( x \right) \right)}^{2}}}{{{\left[ g\left( x \right) \right]}^{3}}} \\
\end{align}\]
