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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (8 x^{2} + 2 x - 5)(- 2 x^{2} - 2 x + 3)(- x^{2} + 5 x + 7)\).
Identifying \(\displaystyle f=8 x^{2} + 2 x - 5\) and \(\displaystyle g=\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right)\) and using the product rule with \(\displaystyle f=8 x^{2} + 2 x - 5 \implies f'=16 x + 2\). This leaves g as \(\displaystyle g = \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 2 x^{2} - 2 x + 3 \implies f'=- 4 x - 2\) and \(\displaystyle g=- x^{2} + 5 x + 7 \implies g'=5 - 2 x\). Popping up a level gives \(\displaystyle g'=(- x^{2} + 5 x + 7)(- 4 x - 2)+(- 2 x^{2} - 2 x + 3)(5 - 2 x)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(8 x^{2} + 2 x - 5)(\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right))+(\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right))(16 x + 2)=\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) \left(8 x^{2} + 2 x - 5\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right) \left(8 x^{2} + 2 x - 5\right) + \left(16 x + 2\right) \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right)\)
\begin{question}Find the derivative of $y = (8 x^{2} + 2 x - 5)(- 2 x^{2} - 2 x + 3)(- x^{2} + 5 x + 7)$.
\soln{9cm}{Identifying $f=8 x^{2} + 2 x - 5$ and $g=\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right)$ and using the product rule with $f=8 x^{2} + 2 x - 5 \implies f'=16 x + 2$. This leaves g as $g = \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 2 x^{2} - 2 x + 3 \implies f'=- 4 x - 2$ and $g=- x^{2} + 5 x + 7 \implies g'=5 - 2 x$. Popping up a level gives $g'=(- x^{2} + 5 x + 7)(- 4 x - 2)+(- 2 x^{2} - 2 x + 3)(5 - 2 x)$Popping up again (Back to the original problem) gives $f'=(8 x^{2} + 2 x - 5)(\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right))+(\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right))(16 x + 2)=\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) \left(8 x^{2} + 2 x - 5\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right) \left(8 x^{2} + 2 x - 5\right) + \left(16 x + 2\right) \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right)$}
\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^{2} + 2 x - 5)(- 2 x^{2} - 2 x + 3)(- x^{2} + 5 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%288%20x%5E%7B2%7D%20%2B%202%20x%20-%205%29%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%29%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%29%20" alt="LaTeX: \displaystyle y = (8 x^{2} + 2 x - 5)(- 2 x^{2} - 2 x + 3)(- x^{2} + 5 x + 7) " data-equation-content=" \displaystyle y = (8 x^{2} + 2 x - 5)(- 2 x^{2} - 2 x + 3)(- x^{2} + 5 x + 7) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=8 x^{2} + 2 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B2%7D%20%2B%202%20x%20-%205%20" alt="LaTeX: \displaystyle f=8 x^{2} + 2 x - 5 " data-equation-content=" \displaystyle f=8 x^{2} + 2 x - 5 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " data-equation-content=" \displaystyle g=\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=8 x^{2} + 2 x - 5 \implies f'=16 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B2%7D%20%2B%202%20x%20-%205%20%5Cimplies%20f%27%3D16%20x%20%2B%202%20" alt="LaTeX: \displaystyle f=8 x^{2} + 2 x - 5 \implies f'=16 x + 2 " data-equation-content=" \displaystyle f=8 x^{2} + 2 x - 5 \implies f'=16 x + 2 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " data-equation-content=" \displaystyle g = \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\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 + 3 \implies f'=- 4 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%20%5Cimplies%20f%27%3D-%204%20x%20-%202%20" alt="LaTeX: \displaystyle f=- 2 x^{2} - 2 x + 3 \implies f'=- 4 x - 2 " data-equation-content=" \displaystyle f=- 2 x^{2} - 2 x + 3 \implies f'=- 4 x - 2 " /> and <img class="equation_image" title=" \displaystyle g=- x^{2} + 5 x + 7 \implies g'=5 - 2 x " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%20%5Cimplies%20g%27%3D5%20-%202%20x%20" alt="LaTeX: \displaystyle g=- x^{2} + 5 x + 7 \implies g'=5 - 2 x " data-equation-content=" \displaystyle g=- x^{2} + 5 x + 7 \implies g'=5 - 2 x " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- x^{2} + 5 x + 7)(- 4 x - 2)+(- 2 x^{2} - 2 x + 3)(5 - 2 x) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%29%28-%204%20x%20-%202%29%2B%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%29%285%20-%202%20x%29%20" alt="LaTeX: \displaystyle g'=(- x^{2} + 5 x + 7)(- 4 x - 2)+(- 2 x^{2} - 2 x + 3)(5 - 2 x) " data-equation-content=" \displaystyle g'=(- x^{2} + 5 x + 7)(- 4 x - 2)+(- 2 x^{2} - 2 x + 3)(5 - 2 x) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(8 x^{2} + 2 x - 5)(\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right))+(\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right))(16 x + 2)=\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) \left(8 x^{2} + 2 x - 5\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right) \left(8 x^{2} + 2 x - 5\right) + \left(16 x + 2\right) \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%288%20x%5E%7B2%7D%20%2B%202%20x%20-%205%29%28%5Cleft%285%20-%202%20x%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%28-%204%20x%20-%202%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%29%2B%28%5Cleft%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%29%2816%20x%20%2B%202%29%3D%5Cleft%285%20-%202%20x%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%202%20x%20-%205%5Cright%29%20%2B%20%5Cleft%28-%204%20x%20-%202%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%202%20x%20-%205%5Cright%29%20%2B%20%5Cleft%2816%20x%20%2B%202%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20-%202%20x%20%2B%203%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle f'=(8 x^{2} + 2 x - 5)(\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right))+(\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right))(16 x + 2)=\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) \left(8 x^{2} + 2 x - 5\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right) \left(8 x^{2} + 2 x - 5\right) + \left(16 x + 2\right) \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " data-equation-content=" \displaystyle f'=(8 x^{2} + 2 x - 5)(\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right))+(\left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right))(16 x + 2)=\left(5 - 2 x\right) \left(- 2 x^{2} - 2 x + 3\right) \left(8 x^{2} + 2 x - 5\right) + \left(- 4 x - 2\right) \left(- x^{2} + 5 x + 7\right) \left(8 x^{2} + 2 x - 5\right) + \left(16 x + 2\right) \left(- 2 x^{2} - 2 x + 3\right) \left(- x^{2} + 5 x + 7\right) " /> </p> </p>