Answer
\[f\left( x \right)g''\left( x \right)+g\left( x \right)f''\left( x \right)+2f'\left( x \right)g'\left( x \right)\]
Work Step by Step
\[\begin{align}
& \frac{d}{dx}\left[ f\left( x \right)g\left( x \right) \right] \\
& \text{Using the product rule} \\
& \frac{d}{dx}\left[ f\left( x \right)g\left( x \right) \right]=f\left( x \right)\frac{d}{dx}\left[ g\left( x \right) \right]+g\left( x \right)\frac{d}{dx}\left[ f\left( x \right) \right] \\
& \frac{d}{dx}\left[ f\left( x \right)g\left( x \right) \right]=f\left( x \right)g'\left( x \right)+g\left( x \right)f'\left( x \right) \\
& \text{Calculate the second derivative} \\
& \frac{{{d}^{2}}}{d{{x}^{2}}}\left[ f\left( x \right)g\left( x \right) \right]=\frac{d}{dx}\left[ f\left( x \right)g'\left( x \right) \right]+\frac{d}{dx}\left[ g\left( x \right)f'\left( x \right) \right] \\
& \text{Using the product rule} \\
& =f\left( x \right)\frac{d}{dx}\left[ g'\left( x \right) \right]+g'\left( x \right)\frac{d}{dx}\left[ f\left( x \right) \right]+g\left( x \right)\frac{d}{dx}\left[ f'\left( x \right) \right] \\
& \text{ }+f'\left( x \right)\frac{d}{dx}\left[ g\left( x \right) \right] \\
& =f\left( x \right)g''\left( x \right)+g'\left( x \right)f'\left( x \right)+g\left( x \right)f''\left( x \right)+f'\left( x \right)g'\left( x \right) \\
& \text{Simplifying} \\
& \frac{{{d}^{2}}}{d{{x}^{2}}}\left[ f\left( x \right)g\left( x \right) \right]=f\left( x \right)g''\left( x \right)+g\left( x \right)f''\left( x \right)+2f'\left( x \right)g'\left( x \right) \\
\end{align}\]
