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Identifying \(\displaystyle f=e^{x}\) and \(\displaystyle g=\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right)\) and using the product rule with \(\displaystyle f=e^{x} \implies f'=e^{x}\). This leaves g as \(\displaystyle g = \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x^{2} + 9 x + 2 \implies f'=18 x + 9\) and \(\displaystyle g=6 x^{2} - 6 x - 9 \implies g'=12 x - 6\). Popping up a level gives \(\displaystyle g'=(6 x^{2} - 6 x - 9)(18 x + 9)+(9 x^{2} + 9 x + 2)(12 x - 6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(e^{x})(\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right))+(\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right))(e^{x})=\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) e^{x} + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right) e^{x} + \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) e^{x}\)

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\begin{question}Find the derivative of $y = (e^{x})(9 x^{2} + 9 x + 2)(6 x^{2} - 6 x - 9)$.
    \soln{9cm}{Identifying $f=e^{x}$ and $g=\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right)$ and using the product rule with $f=e^{x} \implies f'=e^{x}$. This leaves g as $g = \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x^{2} + 9 x + 2 \implies f'=18 x + 9$ and $g=6 x^{2} - 6 x - 9 \implies g'=12 x - 6$. Popping up a level gives $g'=(6 x^{2} - 6 x - 9)(18 x + 9)+(9 x^{2} + 9 x + 2)(12 x - 6)$Popping up again (Back to the original problem) gives $f'=(e^{x})(\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right))+(\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right))(e^{x})=\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) e^{x} + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right) e^{x} + \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) e^{x}$}

\end{question}

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (e^{x})(9 x^{2} + 9 x + 2)(6 x^{2} - 6 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28e%5E%7Bx%7D%29%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%29%286%20x%5E%7B2%7D%20-%206%20x%20-%209%29%20" alt="LaTeX:  \displaystyle y = (e^{x})(9 x^{2} + 9 x + 2)(6 x^{2} - 6 x - 9) " data-equation-content=" \displaystyle y = (e^{x})(9 x^{2} + 9 x + 2)(6 x^{2} - 6 x - 9) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f=e^{x} " data-equation-content=" \displaystyle f=e^{x} " />  and  <img class="equation_image" title=" \displaystyle g=\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%286%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) " data-equation-content=" \displaystyle g=\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=e^{x} \implies f'=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20%5Cimplies%20f%27%3De%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f=e^{x} \implies f'=e^{x} " data-equation-content=" \displaystyle f=e^{x} \implies f'=e^{x} " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%286%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) " data-equation-content=" \displaystyle g = \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=9 x^{2} + 9 x + 2 \implies f'=18 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%20%5Cimplies%20f%27%3D18%20x%20%2B%209%20" alt="LaTeX:  \displaystyle f=9 x^{2} + 9 x + 2 \implies f'=18 x + 9 " data-equation-content=" \displaystyle f=9 x^{2} + 9 x + 2 \implies f'=18 x + 9 " />  and  <img class="equation_image" title=" \displaystyle g=6 x^{2} - 6 x - 9 \implies g'=12 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D6%20x%5E%7B2%7D%20-%206%20x%20-%209%20%5Cimplies%20g%27%3D12%20x%20-%206%20" alt="LaTeX:  \displaystyle g=6 x^{2} - 6 x - 9 \implies g'=12 x - 6 " data-equation-content=" \displaystyle g=6 x^{2} - 6 x - 9 \implies g'=12 x - 6 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(6 x^{2} - 6 x - 9)(18 x + 9)+(9 x^{2} + 9 x + 2)(12 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%286%20x%5E%7B2%7D%20-%206%20x%20-%209%29%2818%20x%20%2B%209%29%2B%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%29%2812%20x%20-%206%29%20" alt="LaTeX:  \displaystyle g'=(6 x^{2} - 6 x - 9)(18 x + 9)+(9 x^{2} + 9 x + 2)(12 x - 6) " data-equation-content=" \displaystyle g'=(6 x^{2} - 6 x - 9)(18 x + 9)+(9 x^{2} + 9 x + 2)(12 x - 6) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(e^{x})(\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right))+(\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right))(e^{x})=\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) e^{x} + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right) e^{x} + \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28e%5E%7Bx%7D%29%28%5Cleft%2812%20x%20-%206%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%5Cright%29%20%2B%20%5Cleft%2818%20x%20%2B%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%29%2B%28%5Cleft%286%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%5Cright%29%29%28e%5E%7Bx%7D%29%3D%5Cleft%2812%20x%20-%206%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%2818%20x%20%2B%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%286%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%202%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(e^{x})(\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right))+(\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right))(e^{x})=\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) e^{x} + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right) e^{x} + \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) e^{x} " data-equation-content=" \displaystyle f'=(e^{x})(\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right))+(\left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right))(e^{x})=\left(12 x - 6\right) \left(9 x^{2} + 9 x + 2\right) e^{x} + \left(18 x + 9\right) \left(6 x^{2} - 6 x - 9\right) e^{x} + \left(6 x^{2} - 6 x - 9\right) \left(9 x^{2} + 9 x + 2\right) e^{x} " /> </p> </p>