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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 9 x^{2} + x + 9)(5 x^{2} + 3 x - 1)(- 9 x^{2} + 4 x + 8)\).
Identifying \(\displaystyle f=- 9 x^{2} + x + 9\) and \(\displaystyle g=\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right)\) and using the product rule with \(\displaystyle f=- 9 x^{2} + x + 9 \implies f'=1 - 18 x\). This leaves g as \(\displaystyle g = \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=5 x^{2} + 3 x - 1 \implies f'=10 x + 3\) and \(\displaystyle g=- 9 x^{2} + 4 x + 8 \implies g'=4 - 18 x\). Popping up a level gives \(\displaystyle g'=(- 9 x^{2} + 4 x + 8)(10 x + 3)+(5 x^{2} + 3 x - 1)(4 - 18 x)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 9 x^{2} + x + 9)(\left(4 - 18 x\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + 4 x + 8\right))+(\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right))(1 - 18 x)=\left(1 - 18 x\right) \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) + \left(4 - 18 x\right) \left(- 9 x^{2} + x + 9\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + x + 9\right) \left(- 9 x^{2} + 4 x + 8\right)\)
\begin{question}Find the derivative of $y = (- 9 x^{2} + x + 9)(5 x^{2} + 3 x - 1)(- 9 x^{2} + 4 x + 8)$.
\soln{9cm}{Identifying $f=- 9 x^{2} + x + 9$ and $g=\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right)$ and using the product rule with $f=- 9 x^{2} + x + 9 \implies f'=1 - 18 x$. This leaves g as $g = \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right)$ which also requires the product rule. Pushing down in the new product rule $f=5 x^{2} + 3 x - 1 \implies f'=10 x + 3$ and $g=- 9 x^{2} + 4 x + 8 \implies g'=4 - 18 x$. Popping up a level gives $g'=(- 9 x^{2} + 4 x + 8)(10 x + 3)+(5 x^{2} + 3 x - 1)(4 - 18 x)$Popping up again (Back to the original problem) gives $f'=(- 9 x^{2} + x + 9)(\left(4 - 18 x\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + 4 x + 8\right))+(\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right))(1 - 18 x)=\left(1 - 18 x\right) \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) + \left(4 - 18 x\right) \left(- 9 x^{2} + x + 9\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + x + 9\right) \left(- 9 x^{2} + 4 x + 8\right)$}
\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 = (- 9 x^{2} + x + 9)(5 x^{2} + 3 x - 1)(- 9 x^{2} + 4 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%209%20x%5E%7B2%7D%20%2B%20x%20%2B%209%29%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%29%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle y = (- 9 x^{2} + x + 9)(5 x^{2} + 3 x - 1)(- 9 x^{2} + 4 x + 8) " data-equation-content=" \displaystyle y = (- 9 x^{2} + x + 9)(5 x^{2} + 3 x - 1)(- 9 x^{2} + 4 x + 8) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 9 x^{2} + x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%209%20x%5E%7B2%7D%20%2B%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=- 9 x^{2} + x + 9 " data-equation-content=" \displaystyle f=- 9 x^{2} + x + 9 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) " data-equation-content=" \displaystyle g=\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 9 x^{2} + x + 9 \implies f'=1 - 18 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%209%20x%5E%7B2%7D%20%2B%20x%20%2B%209%20%5Cimplies%20f%27%3D1%20-%2018%20x%20" alt="LaTeX: \displaystyle f=- 9 x^{2} + x + 9 \implies f'=1 - 18 x " data-equation-content=" \displaystyle f=- 9 x^{2} + x + 9 \implies f'=1 - 18 x " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) " data-equation-content=" \displaystyle g = \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=5 x^{2} + 3 x - 1 \implies f'=10 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20%2B%203%20x%20-%201%20%5Cimplies%20f%27%3D10%20x%20%2B%203%20" alt="LaTeX: \displaystyle f=5 x^{2} + 3 x - 1 \implies f'=10 x + 3 " data-equation-content=" \displaystyle f=5 x^{2} + 3 x - 1 \implies f'=10 x + 3 " /> and <img class="equation_image" title=" \displaystyle g=- 9 x^{2} + 4 x + 8 \implies g'=4 - 18 x " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%20%5Cimplies%20g%27%3D4%20-%2018%20x%20" alt="LaTeX: \displaystyle g=- 9 x^{2} + 4 x + 8 \implies g'=4 - 18 x " data-equation-content=" \displaystyle g=- 9 x^{2} + 4 x + 8 \implies g'=4 - 18 x " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 9 x^{2} + 4 x + 8)(10 x + 3)+(5 x^{2} + 3 x - 1)(4 - 18 x) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%29%2810%20x%20%2B%203%29%2B%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%29%284%20-%2018%20x%29%20" alt="LaTeX: \displaystyle g'=(- 9 x^{2} + 4 x + 8)(10 x + 3)+(5 x^{2} + 3 x - 1)(4 - 18 x) " data-equation-content=" \displaystyle g'=(- 9 x^{2} + 4 x + 8)(10 x + 3)+(5 x^{2} + 3 x - 1)(4 - 18 x) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 9 x^{2} + x + 9)(\left(4 - 18 x\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + 4 x + 8\right))+(\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right))(1 - 18 x)=\left(1 - 18 x\right) \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) + \left(4 - 18 x\right) \left(- 9 x^{2} + x + 9\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + x + 9\right) \left(- 9 x^{2} + 4 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%209%20x%5E%7B2%7D%20%2B%20x%20%2B%209%29%28%5Cleft%284%20-%2018%20x%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%5Cright%29%20%2B%20%5Cleft%2810%20x%20%2B%203%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%5Cright%29%29%2B%28%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%5Cright%29%29%281%20-%2018%20x%29%3D%5Cleft%281%20-%2018%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%5Cright%29%20%2B%20%5Cleft%284%20-%2018%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%20x%20%2B%209%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%203%20x%20-%201%5Cright%29%20%2B%20%5Cleft%2810%20x%20%2B%203%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%20x%20%2B%209%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%204%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 9 x^{2} + x + 9)(\left(4 - 18 x\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + 4 x + 8\right))+(\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right))(1 - 18 x)=\left(1 - 18 x\right) \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) + \left(4 - 18 x\right) \left(- 9 x^{2} + x + 9\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + x + 9\right) \left(- 9 x^{2} + 4 x + 8\right) " data-equation-content=" \displaystyle f'=(- 9 x^{2} + x + 9)(\left(4 - 18 x\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + 4 x + 8\right))+(\left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right))(1 - 18 x)=\left(1 - 18 x\right) \left(- 9 x^{2} + 4 x + 8\right) \left(5 x^{2} + 3 x - 1\right) + \left(4 - 18 x\right) \left(- 9 x^{2} + x + 9\right) \left(5 x^{2} + 3 x - 1\right) + \left(10 x + 3\right) \left(- 9 x^{2} + x + 9\right) \left(- 9 x^{2} + 4 x + 8\right) " /> </p> </p>