Answer
\[\begin{align}
& \text{a}\text{. }f\left( x \right)g\left( x \right)h'\left( x \right)+f\left( x \right)h\left( x \right)g'\left( x \right)+h\left( x \right)g\left( x \right)f'\left( x \right) \\
& \text{b}\text{. }2{{e}^{2x}}\left( {{x}^{2}}+3x-2 \right) \\
\end{align}\]
Work Step by Step
\[\begin{align}
& \text{Let }\frac{d}{dx}\left[ f\left( x \right)g\left( x \right)h\left( x \right) \right] \\
& \text{Write as }\frac{d}{dx}\left[ \left( f\left( x \right)g\left( x \right) \right)h\left( x \right) \right] \\
& \text{a}\text{. Compute the derivative using the product rule} \\
& =f\left( x \right)g\left( x \right)\frac{d}{dx}\left[ h\left( x \right) \right]+h\left( x \right)\frac{d}{dx}\left[ f\left( x \right)g\left( x \right) \right] \\
& =f\left( x \right)g\left( x \right)h'\left( x \right)+h\left( x \right)f\left( x \right)\frac{d}{dx}\left[ g\left( x \right) \right]+h\left( x \right)g\left( x \right)\frac{d}{dx}\left[ f\left( x \right) \right] \\
& =f\left( x \right)g\left( x \right)h'\left( x \right)+h\left( x \right)f\left( x \right)g'\left( x \right)+h(x)g\left( x \right)f'\left( x \right) \\
& \text{Rearranging} \\
& =f\left( x \right)g\left( x \right)h'\left( x \right)+f\left( x \right)h\left( x \right)g'\left( x \right)+h\left( x \right)g\left( x \right)f'\left( x \right) \\
& \\
& \text{b}\text{. }\frac{d}{dx}\left[ {{e}^{2x}}\left( x-1 \right)\left( x+3 \right) \right] \\
& \text{Let }f\left( x \right)={{e}^{2x}}\Rightarrow \text{ }f'\left( x \right)=2{{e}^{2x}} \\
& \text{ }g\left( x \right)=x-1\Rightarrow \text{ }g'\left( x \right)=1 \\
& \text{ }h\left( x \right)=x+3\Rightarrow \text{ }h'\left( x \right)=1 \\
& \text{Substituting into} \\
& =f\left( x \right)g\left( x \right)h'\left( x \right)+f\left( x \right)h\left( x \right)g'\left( x \right)+h\left( x \right)g\left( x \right)f'\left( x \right) \\
& \text{We obtain} \\
& ={{e}^{2x}}\left( x-1 \right)\left( 1 \right)+{{e}^{2x}}\left( x+3 \right)\left( 1 \right)+\left( x-1 \right)\left( x+3 \right)\left( 2{{e}^{2x}} \right) \\
& \text{Simplifying} \\
& =x{{e}^{2x}}-{{e}^{2x}}+x{{e}^{2x}}+3{{e}^{2x}}+\left( {{x}^{2}}+2x-3 \right)\left( 2{{e}^{2x}} \right) \\
& =2x{{e}^{2x}}+2{{e}^{2x}}+2{{x}^{2}}{{e}^{2x}}+4x{{e}^{2x}}-6{{e}^{2x}} \\
& =2{{x}^{2}}{{e}^{2x}}+6x{{e}^{2x}}-4{{e}^{2x}} \\
& \text{Factoring} \\
& =2{{e}^{2x}}\left( {{x}^{2}}+3x-2 \right) \\
\end{align}\]
