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Identifying \(\displaystyle f=- 4 x^{2} - 6 x - 4\) and \(\displaystyle g=\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right)\) and using the product rule with \(\displaystyle f=- 4 x^{2} - 6 x - 4 \implies f'=- 8 x - 6\). This leaves g as \(\displaystyle g = \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=2 x^{2} + x + 8 \implies f'=4 x + 1\) and \(\displaystyle g=9 x^{2} + 8 x - 3 \implies g'=18 x + 8\). Popping up a level gives \(\displaystyle g'=(9 x^{2} + 8 x - 3)(4 x + 1)+(2 x^{2} + x + 8)(18 x + 8)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 4 x^{2} - 6 x - 4)(\left(4 x + 1\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(2 x^{2} + x + 8\right))+(\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right))(- 8 x - 6)=\left(- 8 x - 6\right) \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) + \left(4 x + 1\right) \left(- 4 x^{2} - 6 x - 4\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(- 4 x^{2} - 6 x - 4\right) \left(2 x^{2} + x + 8\right)\)

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\begin{question}Find the derivative of $y = (- 4 x^{2} - 6 x - 4)(2 x^{2} + x + 8)(9 x^{2} + 8 x - 3)$.
    \soln{9cm}{Identifying $f=- 4 x^{2} - 6 x - 4$ and $g=\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right)$ and using the product rule with $f=- 4 x^{2} - 6 x - 4 \implies f'=- 8 x - 6$. This leaves g as $g = \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right)$ which also requires the product rule. Pushing down in the new product rule $f=2 x^{2} + x + 8 \implies f'=4 x + 1$ and $g=9 x^{2} + 8 x - 3 \implies g'=18 x + 8$. Popping up a level gives $g'=(9 x^{2} + 8 x - 3)(4 x + 1)+(2 x^{2} + x + 8)(18 x + 8)$Popping up again (Back to the original problem) gives $f'=(- 4 x^{2} - 6 x - 4)(\left(4 x + 1\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(2 x^{2} + x + 8\right))+(\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right))(- 8 x - 6)=\left(- 8 x - 6\right) \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) + \left(4 x + 1\right) \left(- 4 x^{2} - 6 x - 4\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(- 4 x^{2} - 6 x - 4\right) \left(2 x^{2} + x + 8\right)$}

\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 = (- 4 x^{2} - 6 x - 4)(2 x^{2} + x + 8)(9 x^{2} + 8 x - 3) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%204%20x%5E%7B2%7D%20-%206%20x%20-%204%29%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%29%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%29%20" alt="LaTeX:  \displaystyle y = (- 4 x^{2} - 6 x - 4)(2 x^{2} + x + 8)(9 x^{2} + 8 x - 3) " data-equation-content=" \displaystyle y = (- 4 x^{2} - 6 x - 4)(2 x^{2} + x + 8)(9 x^{2} + 8 x - 3) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 4 x^{2} - 6 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%5E%7B2%7D%20-%206%20x%20-%204%20" alt="LaTeX:  \displaystyle f=- 4 x^{2} - 6 x - 4 " data-equation-content=" \displaystyle f=- 4 x^{2} - 6 x - 4 " />  and  <img class="equation_image" title=" \displaystyle g=\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) " data-equation-content=" \displaystyle g=\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 4 x^{2} - 6 x - 4 \implies f'=- 8 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%5E%7B2%7D%20-%206%20x%20-%204%20%5Cimplies%20f%27%3D-%208%20x%20-%206%20" alt="LaTeX:  \displaystyle f=- 4 x^{2} - 6 x - 4 \implies f'=- 8 x - 6 " data-equation-content=" \displaystyle f=- 4 x^{2} - 6 x - 4 \implies f'=- 8 x - 6 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) " data-equation-content=" \displaystyle g = \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=2 x^{2} + x + 8 \implies f'=4 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20x%5E%7B2%7D%20%2B%20x%20%2B%208%20%5Cimplies%20f%27%3D4%20x%20%2B%201%20" alt="LaTeX:  \displaystyle f=2 x^{2} + x + 8 \implies f'=4 x + 1 " data-equation-content=" \displaystyle f=2 x^{2} + x + 8 \implies f'=4 x + 1 " />  and  <img class="equation_image" title=" \displaystyle g=9 x^{2} + 8 x - 3 \implies g'=18 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20g%3D9%20x%5E%7B2%7D%20%2B%208%20x%20-%203%20%5Cimplies%20g%27%3D18%20x%20%2B%208%20" alt="LaTeX:  \displaystyle g=9 x^{2} + 8 x - 3 \implies g'=18 x + 8 " data-equation-content=" \displaystyle g=9 x^{2} + 8 x - 3 \implies g'=18 x + 8 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(9 x^{2} + 8 x - 3)(4 x + 1)+(2 x^{2} + x + 8)(18 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%29%284%20x%20%2B%201%29%2B%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%29%2818%20x%20%2B%208%29%20" alt="LaTeX:  \displaystyle g'=(9 x^{2} + 8 x - 3)(4 x + 1)+(2 x^{2} + x + 8)(18 x + 8) " data-equation-content=" \displaystyle g'=(9 x^{2} + 8 x - 3)(4 x + 1)+(2 x^{2} + x + 8)(18 x + 8) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 4 x^{2} - 6 x - 4)(\left(4 x + 1\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(2 x^{2} + x + 8\right))+(\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right))(- 8 x - 6)=\left(- 8 x - 6\right) \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) + \left(4 x + 1\right) \left(- 4 x^{2} - 6 x - 4\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(- 4 x^{2} - 6 x - 4\right) \left(2 x^{2} + x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%204%20x%5E%7B2%7D%20-%206%20x%20-%204%29%28%5Cleft%284%20x%20%2B%201%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%2B%20%5Cleft%2818%20x%20%2B%208%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%5Cright%29%29%2B%28%5Cleft%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%29%28-%208%20x%20-%206%29%3D%5Cleft%28-%208%20x%20-%206%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%2B%20%5Cleft%284%20x%20%2B%201%5Cright%29%20%5Cleft%28-%204%20x%5E%7B2%7D%20-%206%20x%20-%204%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%2B%20%5Cleft%2818%20x%20%2B%208%5Cright%29%20%5Cleft%28-%204%20x%5E%7B2%7D%20-%206%20x%20-%204%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%20x%20%2B%208%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(- 4 x^{2} - 6 x - 4)(\left(4 x + 1\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(2 x^{2} + x + 8\right))+(\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right))(- 8 x - 6)=\left(- 8 x - 6\right) \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) + \left(4 x + 1\right) \left(- 4 x^{2} - 6 x - 4\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(- 4 x^{2} - 6 x - 4\right) \left(2 x^{2} + x + 8\right) " data-equation-content=" \displaystyle f'=(- 4 x^{2} - 6 x - 4)(\left(4 x + 1\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(2 x^{2} + x + 8\right))+(\left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right))(- 8 x - 6)=\left(- 8 x - 6\right) \left(2 x^{2} + x + 8\right) \left(9 x^{2} + 8 x - 3\right) + \left(4 x + 1\right) \left(- 4 x^{2} - 6 x - 4\right) \left(9 x^{2} + 8 x - 3\right) + \left(18 x + 8\right) \left(- 4 x^{2} - 6 x - 4\right) \left(2 x^{2} + x + 8\right) " /> </p> </p>