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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 7 x^{2} - 4 x - 4)(- 2 x^{2} + 7 x + 9)(9 x^{2} + 8 x - 8)\).
Identifying \(\displaystyle f=- 7 x^{2} - 4 x - 4\) and \(\displaystyle g=\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right)\) and using the product rule with \(\displaystyle f=- 7 x^{2} - 4 x - 4 \implies f'=- 14 x - 4\). This leaves g as \(\displaystyle g = \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 2 x^{2} + 7 x + 9 \implies f'=7 - 4 x\) and \(\displaystyle g=9 x^{2} + 8 x - 8 \implies g'=18 x + 8\). Popping up a level gives \(\displaystyle g'=(9 x^{2} + 8 x - 8)(7 - 4 x)+(- 2 x^{2} + 7 x + 9)(18 x + 8)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 7 x^{2} - 4 x - 4)(\left(7 - 4 x\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 2 x^{2} + 7 x + 9\right))+(\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right))(- 14 x - 4)=\left(7 - 4 x\right) \left(- 7 x^{2} - 4 x - 4\right) \left(9 x^{2} + 8 x - 8\right) + \left(- 14 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 7 x^{2} - 4 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right)\)
\begin{question}Find the derivative of $y = (- 7 x^{2} - 4 x - 4)(- 2 x^{2} + 7 x + 9)(9 x^{2} + 8 x - 8)$.
\soln{9cm}{Identifying $f=- 7 x^{2} - 4 x - 4$ and $g=\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right)$ and using the product rule with $f=- 7 x^{2} - 4 x - 4 \implies f'=- 14 x - 4$. This leaves g as $g = \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 2 x^{2} + 7 x + 9 \implies f'=7 - 4 x$ and $g=9 x^{2} + 8 x - 8 \implies g'=18 x + 8$. Popping up a level gives $g'=(9 x^{2} + 8 x - 8)(7 - 4 x)+(- 2 x^{2} + 7 x + 9)(18 x + 8)$Popping up again (Back to the original problem) gives $f'=(- 7 x^{2} - 4 x - 4)(\left(7 - 4 x\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 2 x^{2} + 7 x + 9\right))+(\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right))(- 14 x - 4)=\left(7 - 4 x\right) \left(- 7 x^{2} - 4 x - 4\right) \left(9 x^{2} + 8 x - 8\right) + \left(- 14 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 7 x^{2} - 4 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right)$}
\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}<p> <p>Find the derivative of <img class="equation_image" title=" \displaystyle y = (- 7 x^{2} - 4 x - 4)(- 2 x^{2} + 7 x + 9)(9 x^{2} + 8 x - 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%207%20x%5E%7B2%7D%20-%204%20x%20-%204%29%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%29%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%29%20" alt="LaTeX: \displaystyle y = (- 7 x^{2} - 4 x - 4)(- 2 x^{2} + 7 x + 9)(9 x^{2} + 8 x - 8) " data-equation-content=" \displaystyle y = (- 7 x^{2} - 4 x - 4)(- 2 x^{2} + 7 x + 9)(9 x^{2} + 8 x - 8) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 7 x^{2} - 4 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%5E%7B2%7D%20-%204%20x%20-%204%20" alt="LaTeX: \displaystyle f=- 7 x^{2} - 4 x - 4 " data-equation-content=" \displaystyle f=- 7 x^{2} - 4 x - 4 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) " data-equation-content=" \displaystyle g=\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 7 x^{2} - 4 x - 4 \implies f'=- 14 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%5E%7B2%7D%20-%204%20x%20-%204%20%5Cimplies%20f%27%3D-%2014%20x%20-%204%20" alt="LaTeX: \displaystyle f=- 7 x^{2} - 4 x - 4 \implies f'=- 14 x - 4 " data-equation-content=" \displaystyle f=- 7 x^{2} - 4 x - 4 \implies f'=- 14 x - 4 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) " data-equation-content=" \displaystyle g = \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 2 x^{2} + 7 x + 9 \implies f'=7 - 4 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%20%5Cimplies%20f%27%3D7%20-%204%20x%20" alt="LaTeX: \displaystyle f=- 2 x^{2} + 7 x + 9 \implies f'=7 - 4 x " data-equation-content=" \displaystyle f=- 2 x^{2} + 7 x + 9 \implies f'=7 - 4 x " /> and <img class="equation_image" title=" \displaystyle g=9 x^{2} + 8 x - 8 \implies g'=18 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20g%3D9%20x%5E%7B2%7D%20%2B%208%20x%20-%208%20%5Cimplies%20g%27%3D18%20x%20%2B%208%20" alt="LaTeX: \displaystyle g=9 x^{2} + 8 x - 8 \implies g'=18 x + 8 " data-equation-content=" \displaystyle g=9 x^{2} + 8 x - 8 \implies g'=18 x + 8 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(9 x^{2} + 8 x - 8)(7 - 4 x)+(- 2 x^{2} + 7 x + 9)(18 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%29%287%20-%204%20x%29%2B%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%29%2818%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle g'=(9 x^{2} + 8 x - 8)(7 - 4 x)+(- 2 x^{2} + 7 x + 9)(18 x + 8) " data-equation-content=" \displaystyle g'=(9 x^{2} + 8 x - 8)(7 - 4 x)+(- 2 x^{2} + 7 x + 9)(18 x + 8) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 7 x^{2} - 4 x - 4)(\left(7 - 4 x\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 2 x^{2} + 7 x + 9\right))+(\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right))(- 14 x - 4)=\left(7 - 4 x\right) \left(- 7 x^{2} - 4 x - 4\right) \left(9 x^{2} + 8 x - 8\right) + \left(- 14 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 7 x^{2} - 4 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%207%20x%5E%7B2%7D%20-%204%20x%20-%204%29%28%5Cleft%287%20-%204%20x%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%5Cright%29%20%2B%20%5Cleft%2818%20x%20%2B%208%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%5Cright%29%29%2B%28%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%5Cright%29%29%28-%2014%20x%20-%204%29%3D%5Cleft%287%20-%204%20x%5Cright%29%20%5Cleft%28-%207%20x%5E%7B2%7D%20-%204%20x%20-%204%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%5Cright%29%20%2B%20%5Cleft%28-%2014%20x%20-%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%208%20x%20-%208%5Cright%29%20%2B%20%5Cleft%2818%20x%20%2B%208%5Cright%29%20%5Cleft%28-%207%20x%5E%7B2%7D%20-%204%20x%20-%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20%2B%209%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 7 x^{2} - 4 x - 4)(\left(7 - 4 x\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 2 x^{2} + 7 x + 9\right))+(\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right))(- 14 x - 4)=\left(7 - 4 x\right) \left(- 7 x^{2} - 4 x - 4\right) \left(9 x^{2} + 8 x - 8\right) + \left(- 14 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 7 x^{2} - 4 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) " data-equation-content=" \displaystyle f'=(- 7 x^{2} - 4 x - 4)(\left(7 - 4 x\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 2 x^{2} + 7 x + 9\right))+(\left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right))(- 14 x - 4)=\left(7 - 4 x\right) \left(- 7 x^{2} - 4 x - 4\right) \left(9 x^{2} + 8 x - 8\right) + \left(- 14 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) \left(9 x^{2} + 8 x - 8\right) + \left(18 x + 8\right) \left(- 7 x^{2} - 4 x - 4\right) \left(- 2 x^{2} + 7 x + 9\right) " /> </p> </p>