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Identifying \(\displaystyle f=7 x^{2} - 2 x - 2\) and \(\displaystyle g=\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right)\) and using the product rule with \(\displaystyle f=7 x^{2} - 2 x - 2 \implies f'=14 x - 2\). This leaves g as \(\displaystyle g = \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=6 x^{2} - 5 x - 6 \implies f'=12 x - 5\) and \(\displaystyle g=8 x^{2} - 6 x - 1 \implies g'=16 x - 6\). Popping up a level gives \(\displaystyle g'=(8 x^{2} - 6 x - 1)(12 x - 5)+(6 x^{2} - 5 x - 6)(16 x - 6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(7 x^{2} - 2 x - 2)(\left(12 x - 5\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right))+(\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right))(14 x - 2)=\left(12 x - 5\right) \left(7 x^{2} - 2 x - 2\right) \left(8 x^{2} - 6 x - 1\right) + \left(14 x - 2\right) \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right) \left(7 x^{2} - 2 x - 2\right)\)

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\begin{question}Find the derivative of $y = (7 x^{2} - 2 x - 2)(6 x^{2} - 5 x - 6)(8 x^{2} - 6 x - 1)$.
    \soln{9cm}{Identifying $f=7 x^{2} - 2 x - 2$ and $g=\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right)$ and using the product rule with $f=7 x^{2} - 2 x - 2 \implies f'=14 x - 2$. This leaves g as $g = \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right)$ which also requires the product rule. Pushing down in the new product rule $f=6 x^{2} - 5 x - 6 \implies f'=12 x - 5$ and $g=8 x^{2} - 6 x - 1 \implies g'=16 x - 6$. Popping up a level gives $g'=(8 x^{2} - 6 x - 1)(12 x - 5)+(6 x^{2} - 5 x - 6)(16 x - 6)$Popping up again (Back to the original problem) gives $f'=(7 x^{2} - 2 x - 2)(\left(12 x - 5\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right))+(\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right))(14 x - 2)=\left(12 x - 5\right) \left(7 x^{2} - 2 x - 2\right) \left(8 x^{2} - 6 x - 1\right) + \left(14 x - 2\right) \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right) \left(7 x^{2} - 2 x - 2\right)$}

\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 = (7 x^{2} - 2 x - 2)(6 x^{2} - 5 x - 6)(8 x^{2} - 6 x - 1) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%287%20x%5E%7B2%7D%20-%202%20x%20-%202%29%286%20x%5E%7B2%7D%20-%205%20x%20-%206%29%288%20x%5E%7B2%7D%20-%206%20x%20-%201%29%20" alt="LaTeX:  \displaystyle y = (7 x^{2} - 2 x - 2)(6 x^{2} - 5 x - 6)(8 x^{2} - 6 x - 1) " data-equation-content=" \displaystyle y = (7 x^{2} - 2 x - 2)(6 x^{2} - 5 x - 6)(8 x^{2} - 6 x - 1) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=7 x^{2} - 2 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B2%7D%20-%202%20x%20-%202%20" alt="LaTeX:  \displaystyle f=7 x^{2} - 2 x - 2 " data-equation-content=" \displaystyle f=7 x^{2} - 2 x - 2 " />  and  <img class="equation_image" title=" \displaystyle g=\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%286%20x%5E%7B2%7D%20-%205%20x%20-%206%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20-%206%20x%20-%201%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) " data-equation-content=" \displaystyle g=\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=7 x^{2} - 2 x - 2 \implies f'=14 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B2%7D%20-%202%20x%20-%202%20%5Cimplies%20f%27%3D14%20x%20-%202%20" alt="LaTeX:  \displaystyle f=7 x^{2} - 2 x - 2 \implies f'=14 x - 2 " data-equation-content=" \displaystyle f=7 x^{2} - 2 x - 2 \implies f'=14 x - 2 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%286%20x%5E%7B2%7D%20-%205%20x%20-%206%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20-%206%20x%20-%201%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) " data-equation-content=" \displaystyle g = \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=6 x^{2} - 5 x - 6 \implies f'=12 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D6%20x%5E%7B2%7D%20-%205%20x%20-%206%20%5Cimplies%20f%27%3D12%20x%20-%205%20" alt="LaTeX:  \displaystyle f=6 x^{2} - 5 x - 6 \implies f'=12 x - 5 " data-equation-content=" \displaystyle f=6 x^{2} - 5 x - 6 \implies f'=12 x - 5 " />  and  <img class="equation_image" title=" \displaystyle g=8 x^{2} - 6 x - 1 \implies g'=16 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D8%20x%5E%7B2%7D%20-%206%20x%20-%201%20%5Cimplies%20g%27%3D16%20x%20-%206%20" alt="LaTeX:  \displaystyle g=8 x^{2} - 6 x - 1 \implies g'=16 x - 6 " data-equation-content=" \displaystyle g=8 x^{2} - 6 x - 1 \implies g'=16 x - 6 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(8 x^{2} - 6 x - 1)(12 x - 5)+(6 x^{2} - 5 x - 6)(16 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%288%20x%5E%7B2%7D%20-%206%20x%20-%201%29%2812%20x%20-%205%29%2B%286%20x%5E%7B2%7D%20-%205%20x%20-%206%29%2816%20x%20-%206%29%20" alt="LaTeX:  \displaystyle g'=(8 x^{2} - 6 x - 1)(12 x - 5)+(6 x^{2} - 5 x - 6)(16 x - 6) " data-equation-content=" \displaystyle g'=(8 x^{2} - 6 x - 1)(12 x - 5)+(6 x^{2} - 5 x - 6)(16 x - 6) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(7 x^{2} - 2 x - 2)(\left(12 x - 5\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right))+(\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right))(14 x - 2)=\left(12 x - 5\right) \left(7 x^{2} - 2 x - 2\right) \left(8 x^{2} - 6 x - 1\right) + \left(14 x - 2\right) \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right) \left(7 x^{2} - 2 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%287%20x%5E%7B2%7D%20-%202%20x%20-%202%29%28%5Cleft%2812%20x%20-%205%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20-%206%20x%20-%201%5Cright%29%20%2B%20%5Cleft%2816%20x%20-%206%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%205%20x%20-%206%5Cright%29%29%2B%28%5Cleft%286%20x%5E%7B2%7D%20-%205%20x%20-%206%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20-%206%20x%20-%201%5Cright%29%29%2814%20x%20-%202%29%3D%5Cleft%2812%20x%20-%205%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20-%202%20x%20-%202%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20-%206%20x%20-%201%5Cright%29%20%2B%20%5Cleft%2814%20x%20-%202%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%205%20x%20-%206%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20-%206%20x%20-%201%5Cright%29%20%2B%20%5Cleft%2816%20x%20-%206%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%205%20x%20-%206%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20-%202%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(7 x^{2} - 2 x - 2)(\left(12 x - 5\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right))+(\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right))(14 x - 2)=\left(12 x - 5\right) \left(7 x^{2} - 2 x - 2\right) \left(8 x^{2} - 6 x - 1\right) + \left(14 x - 2\right) \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right) \left(7 x^{2} - 2 x - 2\right) " data-equation-content=" \displaystyle f'=(7 x^{2} - 2 x - 2)(\left(12 x - 5\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right))+(\left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right))(14 x - 2)=\left(12 x - 5\right) \left(7 x^{2} - 2 x - 2\right) \left(8 x^{2} - 6 x - 1\right) + \left(14 x - 2\right) \left(6 x^{2} - 5 x - 6\right) \left(8 x^{2} - 6 x - 1\right) + \left(16 x - 6\right) \left(6 x^{2} - 5 x - 6\right) \left(7 x^{2} - 2 x - 2\right) " /> </p> </p>