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Identifying \(\displaystyle f=- 7 x^{2} + 7 x - 4\) and \(\displaystyle g=\left(3 x^{2} + 3 x - 7\right) e^{x}\) and using the product rule with \(\displaystyle f=- 7 x^{2} + 7 x - 4 \implies f'=7 - 14 x\). This leaves g as \(\displaystyle g = \left(3 x^{2} + 3 x - 7\right) e^{x}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=e^{x} \implies f'=e^{x}\) and \(\displaystyle g=3 x^{2} + 3 x - 7 \implies g'=6 x + 3\). Popping up a level gives \(\displaystyle g'=(3 x^{2} + 3 x - 7)(e^{x})+(e^{x})(6 x + 3)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 7 x^{2} + 7 x - 4)(\left(6 x + 3\right) e^{x} + \left(3 x^{2} + 3 x - 7\right) e^{x})+(\left(3 x^{2} + 3 x - 7\right) e^{x})(7 - 14 x)=\left(7 - 14 x\right) \left(3 x^{2} + 3 x - 7\right) e^{x} + \left(6 x + 3\right) \left(- 7 x^{2} + 7 x - 4\right) e^{x} + \left(- 7 x^{2} + 7 x - 4\right) \left(3 x^{2} + 3 x - 7\right) e^{x}\)

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\begin{question}Find the derivative of $y = (- 7 x^{2} + 7 x - 4)(e^{x})(3 x^{2} + 3 x - 7)$.
    \soln{9cm}{Identifying $f=- 7 x^{2} + 7 x - 4$ and $g=\left(3 x^{2} + 3 x - 7\right) e^{x}$ and using the product rule with $f=- 7 x^{2} + 7 x - 4 \implies f'=7 - 14 x$. This leaves g as $g = \left(3 x^{2} + 3 x - 7\right) e^{x}$ which also requires the product rule. Pushing down in the new product rule $f=e^{x} \implies f'=e^{x}$ and $g=3 x^{2} + 3 x - 7 \implies g'=6 x + 3$. Popping up a level gives $g'=(3 x^{2} + 3 x - 7)(e^{x})+(e^{x})(6 x + 3)$Popping up again (Back to the original problem) gives $f'=(- 7 x^{2} + 7 x - 4)(\left(6 x + 3\right) e^{x} + \left(3 x^{2} + 3 x - 7\right) e^{x})+(\left(3 x^{2} + 3 x - 7\right) e^{x})(7 - 14 x)=\left(7 - 14 x\right) \left(3 x^{2} + 3 x - 7\right) e^{x} + \left(6 x + 3\right) \left(- 7 x^{2} + 7 x - 4\right) e^{x} + \left(- 7 x^{2} + 7 x - 4\right) \left(3 x^{2} + 3 x - 7\right) e^{x}$}

\end{question}

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

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HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (- 7 x^{2} + 7 x - 4)(e^{x})(3 x^{2} + 3 x - 7) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%207%20x%5E%7B2%7D%20%2B%207%20x%20-%204%29%28e%5E%7Bx%7D%29%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%29%20" alt="LaTeX:  \displaystyle y = (- 7 x^{2} + 7 x - 4)(e^{x})(3 x^{2} + 3 x - 7) " data-equation-content=" \displaystyle y = (- 7 x^{2} + 7 x - 4)(e^{x})(3 x^{2} + 3 x - 7) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 7 x^{2} + 7 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%5E%7B2%7D%20%2B%207%20x%20-%204%20" alt="LaTeX:  \displaystyle f=- 7 x^{2} + 7 x - 4 " data-equation-content=" \displaystyle f=- 7 x^{2} + 7 x - 4 " />  and  <img class="equation_image" title=" \displaystyle g=\left(3 x^{2} + 3 x - 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g=\left(3 x^{2} + 3 x - 7\right) e^{x} " data-equation-content=" \displaystyle g=\left(3 x^{2} + 3 x - 7\right) e^{x} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 7 x^{2} + 7 x - 4 \implies f'=7 - 14 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%5E%7B2%7D%20%2B%207%20x%20-%204%20%5Cimplies%20f%27%3D7%20-%2014%20x%20" alt="LaTeX:  \displaystyle f=- 7 x^{2} + 7 x - 4 \implies f'=7 - 14 x " data-equation-content=" \displaystyle f=- 7 x^{2} + 7 x - 4 \implies f'=7 - 14 x " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(3 x^{2} + 3 x - 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g = \left(3 x^{2} + 3 x - 7\right) e^{x} " data-equation-content=" \displaystyle g = \left(3 x^{2} + 3 x - 7\right) e^{x} " />  which also requires the product rule. Pushing down in the new product rule  <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} " />  and  <img class="equation_image" title=" \displaystyle g=3 x^{2} + 3 x - 7 \implies g'=6 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D3%20x%5E%7B2%7D%20%2B%203%20x%20-%207%20%5Cimplies%20g%27%3D6%20x%20%2B%203%20" alt="LaTeX:  \displaystyle g=3 x^{2} + 3 x - 7 \implies g'=6 x + 3 " data-equation-content=" \displaystyle g=3 x^{2} + 3 x - 7 \implies g'=6 x + 3 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(3 x^{2} + 3 x - 7)(e^{x})+(e^{x})(6 x + 3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%29%28e%5E%7Bx%7D%29%2B%28e%5E%7Bx%7D%29%286%20x%20%2B%203%29%20" alt="LaTeX:  \displaystyle g'=(3 x^{2} + 3 x - 7)(e^{x})+(e^{x})(6 x + 3) " data-equation-content=" \displaystyle g'=(3 x^{2} + 3 x - 7)(e^{x})+(e^{x})(6 x + 3) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 7 x^{2} + 7 x - 4)(\left(6 x + 3\right) e^{x} + \left(3 x^{2} + 3 x - 7\right) e^{x})+(\left(3 x^{2} + 3 x - 7\right) e^{x})(7 - 14 x)=\left(7 - 14 x\right) \left(3 x^{2} + 3 x - 7\right) e^{x} + \left(6 x + 3\right) \left(- 7 x^{2} + 7 x - 4\right) e^{x} + \left(- 7 x^{2} + 7 x - 4\right) \left(3 x^{2} + 3 x - 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%207%20x%5E%7B2%7D%20%2B%207%20x%20-%204%29%28%5Cleft%286%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%29%2B%28%5Cleft%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%29%287%20-%2014%20x%29%3D%5Cleft%287%20-%2014%20x%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%286%20x%20%2B%203%5Cright%29%20%5Cleft%28-%207%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%207%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20%2B%203%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(- 7 x^{2} + 7 x - 4)(\left(6 x + 3\right) e^{x} + \left(3 x^{2} + 3 x - 7\right) e^{x})+(\left(3 x^{2} + 3 x - 7\right) e^{x})(7 - 14 x)=\left(7 - 14 x\right) \left(3 x^{2} + 3 x - 7\right) e^{x} + \left(6 x + 3\right) \left(- 7 x^{2} + 7 x - 4\right) e^{x} + \left(- 7 x^{2} + 7 x - 4\right) \left(3 x^{2} + 3 x - 7\right) e^{x} " data-equation-content=" \displaystyle f'=(- 7 x^{2} + 7 x - 4)(\left(6 x + 3\right) e^{x} + \left(3 x^{2} + 3 x - 7\right) e^{x})+(\left(3 x^{2} + 3 x - 7\right) e^{x})(7 - 14 x)=\left(7 - 14 x\right) \left(3 x^{2} + 3 x - 7\right) e^{x} + \left(6 x + 3\right) \left(- 7 x^{2} + 7 x - 4\right) e^{x} + \left(- 7 x^{2} + 7 x - 4\right) \left(3 x^{2} + 3 x - 7\right) e^{x} " /> </p> </p>