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Identifying \(\displaystyle f=4 x^{2} + 8 x - 5\) and \(\displaystyle g=\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right)\) and using the product rule with \(\displaystyle f=4 x^{2} + 8 x - 5 \implies f'=8 x + 8\). This leaves g as \(\displaystyle g = \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x^{2} - 3 x - 3 \implies f'=18 x - 3\) and \(\displaystyle g=7 x^{2} + x - 8 \implies g'=14 x + 1\). Popping up a level gives \(\displaystyle g'=(7 x^{2} + x - 8)(18 x - 3)+(9 x^{2} - 3 x - 3)(14 x + 1)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(4 x^{2} + 8 x - 5)(\left(14 x + 1\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(7 x^{2} + x - 8\right))+(\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right))(8 x + 8)=\left(8 x + 8\right) \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) + \left(14 x + 1\right) \left(4 x^{2} + 8 x - 5\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(4 x^{2} + 8 x - 5\right) \left(7 x^{2} + x - 8\right)\)

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\begin{question}Find the derivative of $y = (4 x^{2} + 8 x - 5)(9 x^{2} - 3 x - 3)(7 x^{2} + x - 8)$.
    \soln{9cm}{Identifying $f=4 x^{2} + 8 x - 5$ and $g=\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right)$ and using the product rule with $f=4 x^{2} + 8 x - 5 \implies f'=8 x + 8$. This leaves g as $g = \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x^{2} - 3 x - 3 \implies f'=18 x - 3$ and $g=7 x^{2} + x - 8 \implies g'=14 x + 1$. Popping up a level gives $g'=(7 x^{2} + x - 8)(18 x - 3)+(9 x^{2} - 3 x - 3)(14 x + 1)$Popping up again (Back to the original problem) gives $f'=(4 x^{2} + 8 x - 5)(\left(14 x + 1\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(7 x^{2} + x - 8\right))+(\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right))(8 x + 8)=\left(8 x + 8\right) \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) + \left(14 x + 1\right) \left(4 x^{2} + 8 x - 5\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(4 x^{2} + 8 x - 5\right) \left(7 x^{2} + x - 8\right)$}

\end{question}

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

\end{question}\end{document}

HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (4 x^{2} + 8 x - 5)(9 x^{2} - 3 x - 3)(7 x^{2} + x - 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%284%20x%5E%7B2%7D%20%2B%208%20x%20-%205%29%289%20x%5E%7B2%7D%20-%203%20x%20-%203%29%287%20x%5E%7B2%7D%20%2B%20x%20-%208%29%20" alt="LaTeX:  \displaystyle y = (4 x^{2} + 8 x - 5)(9 x^{2} - 3 x - 3)(7 x^{2} + x - 8) " data-equation-content=" \displaystyle y = (4 x^{2} + 8 x - 5)(9 x^{2} - 3 x - 3)(7 x^{2} + x - 8) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=4 x^{2} + 8 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20x%5E%7B2%7D%20%2B%208%20x%20-%205%20" alt="LaTeX:  \displaystyle f=4 x^{2} + 8 x - 5 " data-equation-content=" \displaystyle f=4 x^{2} + 8 x - 5 " />  and  <img class="equation_image" title=" \displaystyle g=\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%287%20x%5E%7B2%7D%20%2B%20x%20-%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%203%20x%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) " data-equation-content=" \displaystyle g=\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=4 x^{2} + 8 x - 5 \implies f'=8 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20x%5E%7B2%7D%20%2B%208%20x%20-%205%20%5Cimplies%20f%27%3D8%20x%20%2B%208%20" alt="LaTeX:  \displaystyle f=4 x^{2} + 8 x - 5 \implies f'=8 x + 8 " data-equation-content=" \displaystyle f=4 x^{2} + 8 x - 5 \implies f'=8 x + 8 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%287%20x%5E%7B2%7D%20%2B%20x%20-%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%203%20x%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) " data-equation-content=" \displaystyle g = \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=9 x^{2} - 3 x - 3 \implies f'=18 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20-%203%20x%20-%203%20%5Cimplies%20f%27%3D18%20x%20-%203%20" alt="LaTeX:  \displaystyle f=9 x^{2} - 3 x - 3 \implies f'=18 x - 3 " data-equation-content=" \displaystyle f=9 x^{2} - 3 x - 3 \implies f'=18 x - 3 " />  and  <img class="equation_image" title=" \displaystyle g=7 x^{2} + x - 8 \implies g'=14 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20g%3D7%20x%5E%7B2%7D%20%2B%20x%20-%208%20%5Cimplies%20g%27%3D14%20x%20%2B%201%20" alt="LaTeX:  \displaystyle g=7 x^{2} + x - 8 \implies g'=14 x + 1 " data-equation-content=" \displaystyle g=7 x^{2} + x - 8 \implies g'=14 x + 1 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(7 x^{2} + x - 8)(18 x - 3)+(9 x^{2} - 3 x - 3)(14 x + 1) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%287%20x%5E%7B2%7D%20%2B%20x%20-%208%29%2818%20x%20-%203%29%2B%289%20x%5E%7B2%7D%20-%203%20x%20-%203%29%2814%20x%20%2B%201%29%20" alt="LaTeX:  \displaystyle g'=(7 x^{2} + x - 8)(18 x - 3)+(9 x^{2} - 3 x - 3)(14 x + 1) " data-equation-content=" \displaystyle g'=(7 x^{2} + x - 8)(18 x - 3)+(9 x^{2} - 3 x - 3)(14 x + 1) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(4 x^{2} + 8 x - 5)(\left(14 x + 1\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(7 x^{2} + x - 8\right))+(\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right))(8 x + 8)=\left(8 x + 8\right) \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) + \left(14 x + 1\right) \left(4 x^{2} + 8 x - 5\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(4 x^{2} + 8 x - 5\right) \left(7 x^{2} + x - 8\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%284%20x%5E%7B2%7D%20%2B%208%20x%20-%205%29%28%5Cleft%2814%20x%20%2B%201%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%203%20x%20-%203%5Cright%29%20%2B%20%5Cleft%2818%20x%20-%203%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%20x%20-%208%5Cright%29%29%2B%28%5Cleft%287%20x%5E%7B2%7D%20%2B%20x%20-%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%203%20x%20-%203%5Cright%29%29%288%20x%20%2B%208%29%3D%5Cleft%288%20x%20%2B%208%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%20x%20-%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%203%20x%20-%203%5Cright%29%20%2B%20%5Cleft%2814%20x%20%2B%201%5Cright%29%20%5Cleft%284%20x%5E%7B2%7D%20%2B%208%20x%20-%205%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%203%20x%20-%203%5Cright%29%20%2B%20%5Cleft%2818%20x%20-%203%5Cright%29%20%5Cleft%284%20x%5E%7B2%7D%20%2B%208%20x%20-%205%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%20x%20-%208%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(4 x^{2} + 8 x - 5)(\left(14 x + 1\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(7 x^{2} + x - 8\right))+(\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right))(8 x + 8)=\left(8 x + 8\right) \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) + \left(14 x + 1\right) \left(4 x^{2} + 8 x - 5\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(4 x^{2} + 8 x - 5\right) \left(7 x^{2} + x - 8\right) " data-equation-content=" \displaystyle f'=(4 x^{2} + 8 x - 5)(\left(14 x + 1\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(7 x^{2} + x - 8\right))+(\left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right))(8 x + 8)=\left(8 x + 8\right) \left(7 x^{2} + x - 8\right) \left(9 x^{2} - 3 x - 3\right) + \left(14 x + 1\right) \left(4 x^{2} + 8 x - 5\right) \left(9 x^{2} - 3 x - 3\right) + \left(18 x - 3\right) \left(4 x^{2} + 8 x - 5\right) \left(7 x^{2} + x - 8\right) " /> </p> </p>