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Find the derivative of \(\displaystyle y = (- 7 x^{3} + 8 x^{2} - 8 x - 6)(- 5 x^{3} + 5 x^{2} + 2 x + 6)(2 x^{3} - 9 x^{2} + 7 x - 6)\).
Identifying \(\displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6\) and \(\displaystyle g=\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right)\) and using the product rule with \(\displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 \implies f'=- 21 x^{2} + 16 x - 8\). This leaves g as \(\displaystyle g = \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 5 x^{3} + 5 x^{2} + 2 x + 6 \implies f'=- 15 x^{2} + 10 x + 2\) and \(\displaystyle g=2 x^{3} - 9 x^{2} + 7 x - 6 \implies g'=6 x^{2} - 18 x + 7\). Popping up a level gives \(\displaystyle g'=(2 x^{3} - 9 x^{2} + 7 x - 6)(- 15 x^{2} + 10 x + 2)+(- 5 x^{3} + 5 x^{2} + 2 x + 6)(6 x^{2} - 18 x + 7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 7 x^{3} + 8 x^{2} - 8 x - 6)(\left(- 15 x^{2} + 10 x + 2\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right))+(\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right))(- 21 x^{2} + 16 x - 8)=\left(- 21 x^{2} + 16 x - 8\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(- 15 x^{2} + 10 x + 2\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right)\)
\begin{question}Find the derivative of $y = (- 7 x^{3} + 8 x^{2} - 8 x - 6)(- 5 x^{3} + 5 x^{2} + 2 x + 6)(2 x^{3} - 9 x^{2} + 7 x - 6)$.
\soln{9cm}{Identifying $f=- 7 x^{3} + 8 x^{2} - 8 x - 6$ and $g=\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right)$ and using the product rule with $f=- 7 x^{3} + 8 x^{2} - 8 x - 6 \implies f'=- 21 x^{2} + 16 x - 8$. This leaves g as $g = \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 5 x^{3} + 5 x^{2} + 2 x + 6 \implies f'=- 15 x^{2} + 10 x + 2$ and $g=2 x^{3} - 9 x^{2} + 7 x - 6 \implies g'=6 x^{2} - 18 x + 7$. Popping up a level gives $g'=(2 x^{3} - 9 x^{2} + 7 x - 6)(- 15 x^{2} + 10 x + 2)+(- 5 x^{3} + 5 x^{2} + 2 x + 6)(6 x^{2} - 18 x + 7)$Popping up again (Back to the original problem) gives $f'=(- 7 x^{3} + 8 x^{2} - 8 x - 6)(\left(- 15 x^{2} + 10 x + 2\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right))+(\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right))(- 21 x^{2} + 16 x - 8)=\left(- 21 x^{2} + 16 x - 8\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(- 15 x^{2} + 10 x + 2\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right)$}
\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 = (- 7 x^{3} + 8 x^{2} - 8 x - 6)(- 5 x^{3} + 5 x^{2} + 2 x + 6)(2 x^{3} - 9 x^{2} + 7 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%207%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%208%20x%20-%206%29%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%29%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%29%20" alt="LaTeX: \displaystyle y = (- 7 x^{3} + 8 x^{2} - 8 x - 6)(- 5 x^{3} + 5 x^{2} + 2 x + 6)(2 x^{3} - 9 x^{2} + 7 x - 6) " data-equation-content=" \displaystyle y = (- 7 x^{3} + 8 x^{2} - 8 x - 6)(- 5 x^{3} + 5 x^{2} + 2 x + 6)(2 x^{3} - 9 x^{2} + 7 x - 6) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%208%20x%20-%206%20" alt="LaTeX: \displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 " data-equation-content=" \displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%5Cright%29%20%5Cleft%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) " data-equation-content=" \displaystyle g=\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 \implies f'=- 21 x^{2} + 16 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%208%20x%20-%206%20%5Cimplies%20f%27%3D-%2021%20x%5E%7B2%7D%20%2B%2016%20x%20-%208%20" alt="LaTeX: \displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 \implies f'=- 21 x^{2} + 16 x - 8 " data-equation-content=" \displaystyle f=- 7 x^{3} + 8 x^{2} - 8 x - 6 \implies f'=- 21 x^{2} + 16 x - 8 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%5Cright%29%20%5Cleft%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) " data-equation-content=" \displaystyle g = \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 5 x^{3} + 5 x^{2} + 2 x + 6 \implies f'=- 15 x^{2} + 10 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%20%5Cimplies%20f%27%3D-%2015%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%20" alt="LaTeX: \displaystyle f=- 5 x^{3} + 5 x^{2} + 2 x + 6 \implies f'=- 15 x^{2} + 10 x + 2 " data-equation-content=" \displaystyle f=- 5 x^{3} + 5 x^{2} + 2 x + 6 \implies f'=- 15 x^{2} + 10 x + 2 " /> and <img class="equation_image" title=" \displaystyle g=2 x^{3} - 9 x^{2} + 7 x - 6 \implies g'=6 x^{2} - 18 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D2%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%20%5Cimplies%20g%27%3D6%20x%5E%7B2%7D%20-%2018%20x%20%2B%207%20" alt="LaTeX: \displaystyle g=2 x^{3} - 9 x^{2} + 7 x - 6 \implies g'=6 x^{2} - 18 x + 7 " data-equation-content=" \displaystyle g=2 x^{3} - 9 x^{2} + 7 x - 6 \implies g'=6 x^{2} - 18 x + 7 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(2 x^{3} - 9 x^{2} + 7 x - 6)(- 15 x^{2} + 10 x + 2)+(- 5 x^{3} + 5 x^{2} + 2 x + 6)(6 x^{2} - 18 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%29%28-%2015%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%29%2B%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%29%286%20x%5E%7B2%7D%20-%2018%20x%20%2B%207%29%20" alt="LaTeX: \displaystyle g'=(2 x^{3} - 9 x^{2} + 7 x - 6)(- 15 x^{2} + 10 x + 2)+(- 5 x^{3} + 5 x^{2} + 2 x + 6)(6 x^{2} - 18 x + 7) " data-equation-content=" \displaystyle g'=(2 x^{3} - 9 x^{2} + 7 x - 6)(- 15 x^{2} + 10 x + 2)+(- 5 x^{3} + 5 x^{2} + 2 x + 6)(6 x^{2} - 18 x + 7) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 7 x^{3} + 8 x^{2} - 8 x - 6)(\left(- 15 x^{2} + 10 x + 2\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right))+(\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right))(- 21 x^{2} + 16 x - 8)=\left(- 21 x^{2} + 16 x - 8\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(- 15 x^{2} + 10 x + 2\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%207%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%208%20x%20-%206%29%28%5Cleft%28-%2015%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%5Cright%29%20%5Cleft%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%2B%20%5Cleft%286%20x%5E%7B2%7D%20-%2018%20x%20%2B%207%5Cright%29%20%5Cleft%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%5Cright%29%29%2B%28%5Cleft%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%5Cright%29%20%5Cleft%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%29%28-%2021%20x%5E%7B2%7D%20%2B%2016%20x%20-%208%29%3D%5Cleft%28-%2021%20x%5E%7B2%7D%20%2B%2016%20x%20-%208%5Cright%29%20%5Cleft%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%5Cright%29%20%5Cleft%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%2B%20%5Cleft%28-%2015%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%5Cright%29%20%5Cleft%28-%207%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%208%20x%20-%206%5Cright%29%20%5Cleft%282%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%2B%20%5Cleft%286%20x%5E%7B2%7D%20-%2018%20x%20%2B%207%5Cright%29%20%5Cleft%28-%207%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%208%20x%20-%206%5Cright%29%20%5Cleft%28-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20%2B%206%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 7 x^{3} + 8 x^{2} - 8 x - 6)(\left(- 15 x^{2} + 10 x + 2\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right))+(\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right))(- 21 x^{2} + 16 x - 8)=\left(- 21 x^{2} + 16 x - 8\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(- 15 x^{2} + 10 x + 2\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) " data-equation-content=" \displaystyle f'=(- 7 x^{3} + 8 x^{2} - 8 x - 6)(\left(- 15 x^{2} + 10 x + 2\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right))+(\left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right))(- 21 x^{2} + 16 x - 8)=\left(- 21 x^{2} + 16 x - 8\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(- 15 x^{2} + 10 x + 2\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(2 x^{3} - 9 x^{2} + 7 x - 6\right) + \left(6 x^{2} - 18 x + 7\right) \left(- 7 x^{3} + 8 x^{2} - 8 x - 6\right) \left(- 5 x^{3} + 5 x^{2} + 2 x + 6\right) " /> </p> </p>