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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 5 x^{2} - 7 x + 8)(- 2 x^{2} + 2 x + 4)(8 x^{2} + 3 x - 2)\).
Identifying \(\displaystyle f=- 5 x^{2} - 7 x + 8\) and \(\displaystyle g=\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right)\) and using the product rule with \(\displaystyle f=- 5 x^{2} - 7 x + 8 \implies f'=- 10 x - 7\). This leaves g as \(\displaystyle g = \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 2 x^{2} + 2 x + 4 \implies f'=2 - 4 x\) and \(\displaystyle g=8 x^{2} + 3 x - 2 \implies g'=16 x + 3\). Popping up a level gives \(\displaystyle g'=(8 x^{2} + 3 x - 2)(2 - 4 x)+(- 2 x^{2} + 2 x + 4)(16 x + 3)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 5 x^{2} - 7 x + 8)(\left(2 - 4 x\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 2 x^{2} + 2 x + 4\right))+(\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right))(- 10 x - 7)=\left(2 - 4 x\right) \left(- 5 x^{2} - 7 x + 8\right) \left(8 x^{2} + 3 x - 2\right) + \left(- 10 x - 7\right) \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 5 x^{2} - 7 x + 8\right) \left(- 2 x^{2} + 2 x + 4\right)\)
\begin{question}Find the derivative of $y = (- 5 x^{2} - 7 x + 8)(- 2 x^{2} + 2 x + 4)(8 x^{2} + 3 x - 2)$.
\soln{9cm}{Identifying $f=- 5 x^{2} - 7 x + 8$ and $g=\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right)$ and using the product rule with $f=- 5 x^{2} - 7 x + 8 \implies f'=- 10 x - 7$. This leaves g as $g = \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 2 x^{2} + 2 x + 4 \implies f'=2 - 4 x$ and $g=8 x^{2} + 3 x - 2 \implies g'=16 x + 3$. Popping up a level gives $g'=(8 x^{2} + 3 x - 2)(2 - 4 x)+(- 2 x^{2} + 2 x + 4)(16 x + 3)$Popping up again (Back to the original problem) gives $f'=(- 5 x^{2} - 7 x + 8)(\left(2 - 4 x\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 2 x^{2} + 2 x + 4\right))+(\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right))(- 10 x - 7)=\left(2 - 4 x\right) \left(- 5 x^{2} - 7 x + 8\right) \left(8 x^{2} + 3 x - 2\right) + \left(- 10 x - 7\right) \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 5 x^{2} - 7 x + 8\right) \left(- 2 x^{2} + 2 x + 4\right)$}
\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 = (- 5 x^{2} - 7 x + 8)(- 2 x^{2} + 2 x + 4)(8 x^{2} + 3 x - 2) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%208%29%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%29%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%29%20" alt="LaTeX: \displaystyle y = (- 5 x^{2} - 7 x + 8)(- 2 x^{2} + 2 x + 4)(8 x^{2} + 3 x - 2) " data-equation-content=" \displaystyle y = (- 5 x^{2} - 7 x + 8)(- 2 x^{2} + 2 x + 4)(8 x^{2} + 3 x - 2) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 5 x^{2} - 7 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%208%20" alt="LaTeX: \displaystyle f=- 5 x^{2} - 7 x + 8 " data-equation-content=" \displaystyle f=- 5 x^{2} - 7 x + 8 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) " data-equation-content=" \displaystyle g=\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 5 x^{2} - 7 x + 8 \implies f'=- 10 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%208%20%5Cimplies%20f%27%3D-%2010%20x%20-%207%20" alt="LaTeX: \displaystyle f=- 5 x^{2} - 7 x + 8 \implies f'=- 10 x - 7 " data-equation-content=" \displaystyle f=- 5 x^{2} - 7 x + 8 \implies f'=- 10 x - 7 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) " data-equation-content=" \displaystyle g = \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 2 x^{2} + 2 x + 4 \implies f'=2 - 4 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%20%5Cimplies%20f%27%3D2%20-%204%20x%20" alt="LaTeX: \displaystyle f=- 2 x^{2} + 2 x + 4 \implies f'=2 - 4 x " data-equation-content=" \displaystyle f=- 2 x^{2} + 2 x + 4 \implies f'=2 - 4 x " /> and <img class="equation_image" title=" \displaystyle g=8 x^{2} + 3 x - 2 \implies g'=16 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D8%20x%5E%7B2%7D%20%2B%203%20x%20-%202%20%5Cimplies%20g%27%3D16%20x%20%2B%203%20" alt="LaTeX: \displaystyle g=8 x^{2} + 3 x - 2 \implies g'=16 x + 3 " data-equation-content=" \displaystyle g=8 x^{2} + 3 x - 2 \implies g'=16 x + 3 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(8 x^{2} + 3 x - 2)(2 - 4 x)+(- 2 x^{2} + 2 x + 4)(16 x + 3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%29%282%20-%204%20x%29%2B%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%29%2816%20x%20%2B%203%29%20" alt="LaTeX: \displaystyle g'=(8 x^{2} + 3 x - 2)(2 - 4 x)+(- 2 x^{2} + 2 x + 4)(16 x + 3) " data-equation-content=" \displaystyle g'=(8 x^{2} + 3 x - 2)(2 - 4 x)+(- 2 x^{2} + 2 x + 4)(16 x + 3) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 5 x^{2} - 7 x + 8)(\left(2 - 4 x\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 2 x^{2} + 2 x + 4\right))+(\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right))(- 10 x - 7)=\left(2 - 4 x\right) \left(- 5 x^{2} - 7 x + 8\right) \left(8 x^{2} + 3 x - 2\right) + \left(- 10 x - 7\right) \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 5 x^{2} - 7 x + 8\right) \left(- 2 x^{2} + 2 x + 4\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%208%29%28%5Cleft%282%20-%204%20x%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%2B%20%5Cleft%2816%20x%20%2B%203%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%29%2B%28%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%29%28-%2010%20x%20-%207%29%3D%5Cleft%282%20-%204%20x%5Cright%29%20%5Cleft%28-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%2B%20%5Cleft%28-%2010%20x%20-%207%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%2B%20%5Cleft%2816%20x%20%2B%203%5Cright%29%20%5Cleft%28-%205%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 5 x^{2} - 7 x + 8)(\left(2 - 4 x\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 2 x^{2} + 2 x + 4\right))+(\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right))(- 10 x - 7)=\left(2 - 4 x\right) \left(- 5 x^{2} - 7 x + 8\right) \left(8 x^{2} + 3 x - 2\right) + \left(- 10 x - 7\right) \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 5 x^{2} - 7 x + 8\right) \left(- 2 x^{2} + 2 x + 4\right) " data-equation-content=" \displaystyle f'=(- 5 x^{2} - 7 x + 8)(\left(2 - 4 x\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 2 x^{2} + 2 x + 4\right))+(\left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right))(- 10 x - 7)=\left(2 - 4 x\right) \left(- 5 x^{2} - 7 x + 8\right) \left(8 x^{2} + 3 x - 2\right) + \left(- 10 x - 7\right) \left(- 2 x^{2} + 2 x + 4\right) \left(8 x^{2} + 3 x - 2\right) + \left(16 x + 3\right) \left(- 5 x^{2} - 7 x + 8\right) \left(- 2 x^{2} + 2 x + 4\right) " /> </p> </p>