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Find the derivative of \(\displaystyle y = (- 8 x^{3} + 8 x^{2} - 4 x + 8)(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 2 x^{3} - 9 x^{2} + 4 x + 1)\).
Identifying \(\displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8\) and \(\displaystyle g=\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right)\) and using the product rule with \(\displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 \implies f'=- 24 x^{2} + 16 x - 4\). This leaves g as \(\displaystyle g = \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 3 x^{3} + 4 x^{2} + 8 x - 3 \implies f'=- 9 x^{2} + 8 x + 8\) and \(\displaystyle g=- 2 x^{3} - 9 x^{2} + 4 x + 1 \implies g'=- 6 x^{2} - 18 x + 4\). Popping up a level gives \(\displaystyle g'=(- 2 x^{3} - 9 x^{2} + 4 x + 1)(- 9 x^{2} + 8 x + 8)+(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 6 x^{2} - 18 x + 4)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 8 x^{3} + 8 x^{2} - 4 x + 8)(\left(- 9 x^{2} + 8 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right))+(\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right))(- 24 x^{2} + 16 x - 4)=\left(- 24 x^{2} + 16 x - 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 9 x^{2} + 8 x + 8\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right)\)
\begin{question}Find the derivative of $y = (- 8 x^{3} + 8 x^{2} - 4 x + 8)(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 2 x^{3} - 9 x^{2} + 4 x + 1)$.
\soln{9cm}{Identifying $f=- 8 x^{3} + 8 x^{2} - 4 x + 8$ and $g=\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right)$ and using the product rule with $f=- 8 x^{3} + 8 x^{2} - 4 x + 8 \implies f'=- 24 x^{2} + 16 x - 4$. This leaves g as $g = \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 3 x^{3} + 4 x^{2} + 8 x - 3 \implies f'=- 9 x^{2} + 8 x + 8$ and $g=- 2 x^{3} - 9 x^{2} + 4 x + 1 \implies g'=- 6 x^{2} - 18 x + 4$. Popping up a level gives $g'=(- 2 x^{3} - 9 x^{2} + 4 x + 1)(- 9 x^{2} + 8 x + 8)+(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 6 x^{2} - 18 x + 4)$Popping up again (Back to the original problem) gives $f'=(- 8 x^{3} + 8 x^{2} - 4 x + 8)(\left(- 9 x^{2} + 8 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right))+(\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right))(- 24 x^{2} + 16 x - 4)=\left(- 24 x^{2} + 16 x - 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 9 x^{2} + 8 x + 8\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right)$}
\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}<p> <p>Find the derivative of <img class="equation_image" title=" \displaystyle y = (- 8 x^{3} + 8 x^{2} - 4 x + 8)(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 2 x^{3} - 9 x^{2} + 4 x + 1) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%208%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%204%20x%20%2B%208%29%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%29%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%29%20" alt="LaTeX: \displaystyle y = (- 8 x^{3} + 8 x^{2} - 4 x + 8)(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 2 x^{3} - 9 x^{2} + 4 x + 1) " data-equation-content=" \displaystyle y = (- 8 x^{3} + 8 x^{2} - 4 x + 8)(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 2 x^{3} - 9 x^{2} + 4 x + 1) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%208%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%204%20x%20%2B%208%20" alt="LaTeX: \displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 " data-equation-content=" \displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) " data-equation-content=" \displaystyle g=\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 \implies f'=- 24 x^{2} + 16 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%208%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%204%20x%20%2B%208%20%5Cimplies%20f%27%3D-%2024%20x%5E%7B2%7D%20%2B%2016%20x%20-%204%20" alt="LaTeX: \displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 \implies f'=- 24 x^{2} + 16 x - 4 " data-equation-content=" \displaystyle f=- 8 x^{3} + 8 x^{2} - 4 x + 8 \implies f'=- 24 x^{2} + 16 x - 4 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) " data-equation-content=" \displaystyle g = \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 3 x^{3} + 4 x^{2} + 8 x - 3 \implies f'=- 9 x^{2} + 8 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%20%5Cimplies%20f%27%3D-%209%20x%5E%7B2%7D%20%2B%208%20x%20%2B%208%20" alt="LaTeX: \displaystyle f=- 3 x^{3} + 4 x^{2} + 8 x - 3 \implies f'=- 9 x^{2} + 8 x + 8 " data-equation-content=" \displaystyle f=- 3 x^{3} + 4 x^{2} + 8 x - 3 \implies f'=- 9 x^{2} + 8 x + 8 " /> and <img class="equation_image" title=" \displaystyle g=- 2 x^{3} - 9 x^{2} + 4 x + 1 \implies g'=- 6 x^{2} - 18 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%20%5Cimplies%20g%27%3D-%206%20x%5E%7B2%7D%20-%2018%20x%20%2B%204%20" alt="LaTeX: \displaystyle g=- 2 x^{3} - 9 x^{2} + 4 x + 1 \implies g'=- 6 x^{2} - 18 x + 4 " data-equation-content=" \displaystyle g=- 2 x^{3} - 9 x^{2} + 4 x + 1 \implies g'=- 6 x^{2} - 18 x + 4 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 2 x^{3} - 9 x^{2} + 4 x + 1)(- 9 x^{2} + 8 x + 8)+(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 6 x^{2} - 18 x + 4) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%29%28-%209%20x%5E%7B2%7D%20%2B%208%20x%20%2B%208%29%2B%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%29%28-%206%20x%5E%7B2%7D%20-%2018%20x%20%2B%204%29%20" alt="LaTeX: \displaystyle g'=(- 2 x^{3} - 9 x^{2} + 4 x + 1)(- 9 x^{2} + 8 x + 8)+(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 6 x^{2} - 18 x + 4) " data-equation-content=" \displaystyle g'=(- 2 x^{3} - 9 x^{2} + 4 x + 1)(- 9 x^{2} + 8 x + 8)+(- 3 x^{3} + 4 x^{2} + 8 x - 3)(- 6 x^{2} - 18 x + 4) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 8 x^{3} + 8 x^{2} - 4 x + 8)(\left(- 9 x^{2} + 8 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right))+(\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right))(- 24 x^{2} + 16 x - 4)=\left(- 24 x^{2} + 16 x - 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 9 x^{2} + 8 x + 8\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%208%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%204%20x%20%2B%208%29%28%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%208%20x%20%2B%208%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%5Cright%29%20%2B%20%5Cleft%28-%206%20x%5E%7B2%7D%20-%2018%20x%20%2B%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%29%2B%28%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%5Cright%29%29%28-%2024%20x%5E%7B2%7D%20%2B%2016%20x%20-%204%29%3D%5Cleft%28-%2024%20x%5E%7B2%7D%20%2B%2016%20x%20-%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%5Cright%29%20%2B%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%208%20x%20%2B%208%5Cright%29%20%5Cleft%28-%208%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%204%20x%20%2B%208%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%201%5Cright%29%20%2B%20%5Cleft%28-%206%20x%5E%7B2%7D%20-%2018%20x%20%2B%204%5Cright%29%20%5Cleft%28-%208%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%204%20x%20%2B%208%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%208%20x%20-%203%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 8 x^{3} + 8 x^{2} - 4 x + 8)(\left(- 9 x^{2} + 8 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right))+(\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right))(- 24 x^{2} + 16 x - 4)=\left(- 24 x^{2} + 16 x - 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 9 x^{2} + 8 x + 8\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) " data-equation-content=" \displaystyle f'=(- 8 x^{3} + 8 x^{2} - 4 x + 8)(\left(- 9 x^{2} + 8 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right))+(\left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right))(- 24 x^{2} + 16 x - 4)=\left(- 24 x^{2} + 16 x - 4\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 9 x^{2} + 8 x + 8\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 2 x^{3} - 9 x^{2} + 4 x + 1\right) + \left(- 6 x^{2} - 18 x + 4\right) \left(- 8 x^{3} + 8 x^{2} - 4 x + 8\right) \left(- 3 x^{3} + 4 x^{2} + 8 x - 3\right) " /> </p> </p>