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Identifying \(\displaystyle f=e^{x}\) and \(\displaystyle g=\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right)\) and using the product rule with \(\displaystyle f=e^{x} \implies f'=e^{x}\). This leaves g as \(\displaystyle g = \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 2 x^{2} + 7 x + 7 \implies f'=7 - 4 x\) and \(\displaystyle g=- 9 x^{2} - 9 x + 4 \implies g'=- 18 x - 9\). Popping up a level gives \(\displaystyle g'=(- 9 x^{2} - 9 x + 4)(7 - 4 x)+(- 2 x^{2} + 7 x + 7)(- 18 x - 9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(e^{x})(\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right))+(\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right))(e^{x})=\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) e^{x} + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} + \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x}\)

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\begin{question}Find the derivative of $y = (e^{x})(- 2 x^{2} + 7 x + 7)(- 9 x^{2} - 9 x + 4)$.
    \soln{9cm}{Identifying $f=e^{x}$ and $g=\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right)$ and using the product rule with $f=e^{x} \implies f'=e^{x}$. This leaves g as $g = \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 2 x^{2} + 7 x + 7 \implies f'=7 - 4 x$ and $g=- 9 x^{2} - 9 x + 4 \implies g'=- 18 x - 9$. Popping up a level gives $g'=(- 9 x^{2} - 9 x + 4)(7 - 4 x)+(- 2 x^{2} + 7 x + 7)(- 18 x - 9)$Popping up again (Back to the original problem) gives $f'=(e^{x})(\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right))+(\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right))(e^{x})=\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) e^{x} + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} + \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x}$}

\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})(- 2 x^{2} + 7 x + 7)(- 9 x^{2} - 9 x + 4) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28e%5E%7Bx%7D%29%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%29%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%29%20" alt="LaTeX:  \displaystyle y = (e^{x})(- 2 x^{2} + 7 x + 7)(- 9 x^{2} - 9 x + 4) " data-equation-content=" \displaystyle y = (e^{x})(- 2 x^{2} + 7 x + 7)(- 9 x^{2} - 9 x + 4) " /> .</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(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) " data-equation-content=" \displaystyle g=\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\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(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) " data-equation-content=" \displaystyle g = \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=- 2 x^{2} + 7 x + 7 \implies f'=7 - 4 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%20%5Cimplies%20f%27%3D7%20-%204%20x%20" alt="LaTeX:  \displaystyle f=- 2 x^{2} + 7 x + 7 \implies f'=7 - 4 x " data-equation-content=" \displaystyle f=- 2 x^{2} + 7 x + 7 \implies f'=7 - 4 x " />  and  <img class="equation_image" title=" \displaystyle g=- 9 x^{2} - 9 x + 4 \implies g'=- 18 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%20%5Cimplies%20g%27%3D-%2018%20x%20-%209%20" alt="LaTeX:  \displaystyle g=- 9 x^{2} - 9 x + 4 \implies g'=- 18 x - 9 " data-equation-content=" \displaystyle g=- 9 x^{2} - 9 x + 4 \implies g'=- 18 x - 9 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 9 x^{2} - 9 x + 4)(7 - 4 x)+(- 2 x^{2} + 7 x + 7)(- 18 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%29%287%20-%204%20x%29%2B%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%29%28-%2018%20x%20-%209%29%20" alt="LaTeX:  \displaystyle g'=(- 9 x^{2} - 9 x + 4)(7 - 4 x)+(- 2 x^{2} + 7 x + 7)(- 18 x - 9) " data-equation-content=" \displaystyle g'=(- 9 x^{2} - 9 x + 4)(7 - 4 x)+(- 2 x^{2} + 7 x + 7)(- 18 x - 9) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(e^{x})(\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right))+(\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right))(e^{x})=\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) e^{x} + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} + \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28e%5E%7Bx%7D%29%28%5Cleft%287%20-%204%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%2B%20%5Cleft%28-%2018%20x%20-%209%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%5Cright%29%29%2B%28%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%5Cright%29%29%28e%5E%7Bx%7D%29%3D%5Cleft%287%20-%204%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%2018%20x%20-%209%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(e^{x})(\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right))+(\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right))(e^{x})=\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) e^{x} + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} + \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} " data-equation-content=" \displaystyle f'=(e^{x})(\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right))+(\left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right))(e^{x})=\left(7 - 4 x\right) \left(- 9 x^{2} - 9 x + 4\right) e^{x} + \left(- 18 x - 9\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} + \left(- 9 x^{2} - 9 x + 4\right) \left(- 2 x^{2} + 7 x + 7\right) e^{x} " /> </p> </p>