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Identifying \(\displaystyle f=- 5 x^{2} - 3 x - 8\) and \(\displaystyle g=\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right)\) and using the product rule with \(\displaystyle f=- 5 x^{2} - 3 x - 8 \implies f'=- 10 x - 3\). This leaves g as \(\displaystyle g = \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=6 x^{2} - x - 5 \implies f'=12 x - 1\) and \(\displaystyle g=3 x^{2} - 5 x - 9 \implies g'=6 x - 5\). Popping up a level gives \(\displaystyle g'=(3 x^{2} - 5 x - 9)(12 x - 1)+(6 x^{2} - x - 5)(6 x - 5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 5 x^{2} - 3 x - 8)(\left(6 x - 5\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(3 x^{2} - 5 x - 9\right))+(\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) + \left(6 x - 5\right) \left(- 5 x^{2} - 3 x - 8\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(- 5 x^{2} - 3 x - 8\right) \left(3 x^{2} - 5 x - 9\right)\)

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\begin{question}Find the derivative of $y = (- 5 x^{2} - 3 x - 8)(6 x^{2} - x - 5)(3 x^{2} - 5 x - 9)$.
    \soln{9cm}{Identifying $f=- 5 x^{2} - 3 x - 8$ and $g=\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right)$ and using the product rule with $f=- 5 x^{2} - 3 x - 8 \implies f'=- 10 x - 3$. This leaves g as $g = \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right)$ which also requires the product rule. Pushing down in the new product rule $f=6 x^{2} - x - 5 \implies f'=12 x - 1$ and $g=3 x^{2} - 5 x - 9 \implies g'=6 x - 5$. Popping up a level gives $g'=(3 x^{2} - 5 x - 9)(12 x - 1)+(6 x^{2} - x - 5)(6 x - 5)$Popping up again (Back to the original problem) gives $f'=(- 5 x^{2} - 3 x - 8)(\left(6 x - 5\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(3 x^{2} - 5 x - 9\right))+(\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) + \left(6 x - 5\right) \left(- 5 x^{2} - 3 x - 8\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(- 5 x^{2} - 3 x - 8\right) \left(3 x^{2} - 5 x - 9\right)$}

\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 = (- 5 x^{2} - 3 x - 8)(6 x^{2} - x - 5)(3 x^{2} - 5 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%205%20x%5E%7B2%7D%20-%203%20x%20-%208%29%286%20x%5E%7B2%7D%20-%20x%20-%205%29%283%20x%5E%7B2%7D%20-%205%20x%20-%209%29%20" alt="LaTeX:  \displaystyle y = (- 5 x^{2} - 3 x - 8)(6 x^{2} - x - 5)(3 x^{2} - 5 x - 9) " data-equation-content=" \displaystyle y = (- 5 x^{2} - 3 x - 8)(6 x^{2} - x - 5)(3 x^{2} - 5 x - 9) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 5 x^{2} - 3 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B2%7D%20-%203%20x%20-%208%20" alt="LaTeX:  \displaystyle f=- 5 x^{2} - 3 x - 8 " data-equation-content=" \displaystyle f=- 5 x^{2} - 3 x - 8 " />  and  <img class="equation_image" title=" \displaystyle g=\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%283%20x%5E%7B2%7D%20-%205%20x%20-%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%20x%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) " data-equation-content=" \displaystyle g=\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 5 x^{2} - 3 x - 8 \implies f'=- 10 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B2%7D%20-%203%20x%20-%208%20%5Cimplies%20f%27%3D-%2010%20x%20-%203%20" alt="LaTeX:  \displaystyle f=- 5 x^{2} - 3 x - 8 \implies f'=- 10 x - 3 " data-equation-content=" \displaystyle f=- 5 x^{2} - 3 x - 8 \implies f'=- 10 x - 3 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%283%20x%5E%7B2%7D%20-%205%20x%20-%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%20x%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) " data-equation-content=" \displaystyle g = \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=6 x^{2} - x - 5 \implies f'=12 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D6%20x%5E%7B2%7D%20-%20x%20-%205%20%5Cimplies%20f%27%3D12%20x%20-%201%20" alt="LaTeX:  \displaystyle f=6 x^{2} - x - 5 \implies f'=12 x - 1 " data-equation-content=" \displaystyle f=6 x^{2} - x - 5 \implies f'=12 x - 1 " />  and  <img class="equation_image" title=" \displaystyle g=3 x^{2} - 5 x - 9 \implies g'=6 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D3%20x%5E%7B2%7D%20-%205%20x%20-%209%20%5Cimplies%20g%27%3D6%20x%20-%205%20" alt="LaTeX:  \displaystyle g=3 x^{2} - 5 x - 9 \implies g'=6 x - 5 " data-equation-content=" \displaystyle g=3 x^{2} - 5 x - 9 \implies g'=6 x - 5 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(3 x^{2} - 5 x - 9)(12 x - 1)+(6 x^{2} - x - 5)(6 x - 5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%283%20x%5E%7B2%7D%20-%205%20x%20-%209%29%2812%20x%20-%201%29%2B%286%20x%5E%7B2%7D%20-%20x%20-%205%29%286%20x%20-%205%29%20" alt="LaTeX:  \displaystyle g'=(3 x^{2} - 5 x - 9)(12 x - 1)+(6 x^{2} - x - 5)(6 x - 5) " data-equation-content=" \displaystyle g'=(3 x^{2} - 5 x - 9)(12 x - 1)+(6 x^{2} - x - 5)(6 x - 5) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 5 x^{2} - 3 x - 8)(\left(6 x - 5\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(3 x^{2} - 5 x - 9\right))+(\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) + \left(6 x - 5\right) \left(- 5 x^{2} - 3 x - 8\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(- 5 x^{2} - 3 x - 8\right) \left(3 x^{2} - 5 x - 9\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%205%20x%5E%7B2%7D%20-%203%20x%20-%208%29%28%5Cleft%286%20x%20-%205%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%20x%20-%205%5Cright%29%20%2B%20%5Cleft%2812%20x%20-%201%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20-%205%20x%20-%209%5Cright%29%29%2B%28%5Cleft%283%20x%5E%7B2%7D%20-%205%20x%20-%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%20x%20-%205%5Cright%29%29%28-%2010%20x%20-%203%29%3D%5Cleft%28-%2010%20x%20-%203%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20-%205%20x%20-%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%20x%20-%205%5Cright%29%20%2B%20%5Cleft%286%20x%20-%205%5Cright%29%20%5Cleft%28-%205%20x%5E%7B2%7D%20-%203%20x%20-%208%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%20x%20-%205%5Cright%29%20%2B%20%5Cleft%2812%20x%20-%201%5Cright%29%20%5Cleft%28-%205%20x%5E%7B2%7D%20-%203%20x%20-%208%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20-%205%20x%20-%209%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(- 5 x^{2} - 3 x - 8)(\left(6 x - 5\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(3 x^{2} - 5 x - 9\right))+(\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) + \left(6 x - 5\right) \left(- 5 x^{2} - 3 x - 8\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(- 5 x^{2} - 3 x - 8\right) \left(3 x^{2} - 5 x - 9\right) " data-equation-content=" \displaystyle f'=(- 5 x^{2} - 3 x - 8)(\left(6 x - 5\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(3 x^{2} - 5 x - 9\right))+(\left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(3 x^{2} - 5 x - 9\right) \left(6 x^{2} - x - 5\right) + \left(6 x - 5\right) \left(- 5 x^{2} - 3 x - 8\right) \left(6 x^{2} - x - 5\right) + \left(12 x - 1\right) \left(- 5 x^{2} - 3 x - 8\right) \left(3 x^{2} - 5 x - 9\right) " /> </p> </p>