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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (x^{2} + 3 x + 9)(- 2 x^{2} + 7 x - 8)(2 x^{2} + 5 x - 5)\).
Identifying \(\displaystyle f=x^{2} + 3 x + 9\) and \(\displaystyle g=\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right)\) and using the product rule with \(\displaystyle f=x^{2} + 3 x + 9 \implies f'=2 x + 3\). This leaves g as \(\displaystyle g = \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 2 x^{2} + 7 x - 8 \implies f'=7 - 4 x\) and \(\displaystyle g=2 x^{2} + 5 x - 5 \implies g'=4 x + 5\). Popping up a level gives \(\displaystyle g'=(2 x^{2} + 5 x - 5)(7 - 4 x)+(- 2 x^{2} + 7 x - 8)(4 x + 5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(x^{2} + 3 x + 9)(\left(7 - 4 x\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right))+(\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right))(2 x + 3)=\left(7 - 4 x\right) \left(x^{2} + 3 x + 9\right) \left(2 x^{2} + 5 x - 5\right) + \left(2 x + 3\right) \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right) \left(x^{2} + 3 x + 9\right)\)
\begin{question}Find the derivative of $y = (x^{2} + 3 x + 9)(- 2 x^{2} + 7 x - 8)(2 x^{2} + 5 x - 5)$.
\soln{9cm}{Identifying $f=x^{2} + 3 x + 9$ and $g=\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right)$ and using the product rule with $f=x^{2} + 3 x + 9 \implies f'=2 x + 3$. This leaves g as $g = \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 2 x^{2} + 7 x - 8 \implies f'=7 - 4 x$ and $g=2 x^{2} + 5 x - 5 \implies g'=4 x + 5$. Popping up a level gives $g'=(2 x^{2} + 5 x - 5)(7 - 4 x)+(- 2 x^{2} + 7 x - 8)(4 x + 5)$Popping up again (Back to the original problem) gives $f'=(x^{2} + 3 x + 9)(\left(7 - 4 x\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right))+(\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right))(2 x + 3)=\left(7 - 4 x\right) \left(x^{2} + 3 x + 9\right) \left(2 x^{2} + 5 x - 5\right) + \left(2 x + 3\right) \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right) \left(x^{2} + 3 x + 9\right)$}
\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 = (x^{2} + 3 x + 9)(- 2 x^{2} + 7 x - 8)(2 x^{2} + 5 x - 5) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28x%5E%7B2%7D%20%2B%203%20x%20%2B%209%29%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%29%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%29%20" alt="LaTeX: \displaystyle y = (x^{2} + 3 x + 9)(- 2 x^{2} + 7 x - 8)(2 x^{2} + 5 x - 5) " data-equation-content=" \displaystyle y = (x^{2} + 3 x + 9)(- 2 x^{2} + 7 x - 8)(2 x^{2} + 5 x - 5) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=x^{2} + 3 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3Dx%5E%7B2%7D%20%2B%203%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=x^{2} + 3 x + 9 " data-equation-content=" \displaystyle f=x^{2} + 3 x + 9 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) " data-equation-content=" \displaystyle g=\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=x^{2} + 3 x + 9 \implies f'=2 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3Dx%5E%7B2%7D%20%2B%203%20x%20%2B%209%20%5Cimplies%20f%27%3D2%20x%20%2B%203%20" alt="LaTeX: \displaystyle f=x^{2} + 3 x + 9 \implies f'=2 x + 3 " data-equation-content=" \displaystyle f=x^{2} + 3 x + 9 \implies f'=2 x + 3 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) " data-equation-content=" \displaystyle g = \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\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 - 8 \implies f'=7 - 4 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%20%5Cimplies%20f%27%3D7%20-%204%20x%20" alt="LaTeX: \displaystyle f=- 2 x^{2} + 7 x - 8 \implies f'=7 - 4 x " data-equation-content=" \displaystyle f=- 2 x^{2} + 7 x - 8 \implies f'=7 - 4 x " /> and <img class="equation_image" title=" \displaystyle g=2 x^{2} + 5 x - 5 \implies g'=4 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D2%20x%5E%7B2%7D%20%2B%205%20x%20-%205%20%5Cimplies%20g%27%3D4%20x%20%2B%205%20" alt="LaTeX: \displaystyle g=2 x^{2} + 5 x - 5 \implies g'=4 x + 5 " data-equation-content=" \displaystyle g=2 x^{2} + 5 x - 5 \implies g'=4 x + 5 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(2 x^{2} + 5 x - 5)(7 - 4 x)+(- 2 x^{2} + 7 x - 8)(4 x + 5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%29%287%20-%204%20x%29%2B%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%29%284%20x%20%2B%205%29%20" alt="LaTeX: \displaystyle g'=(2 x^{2} + 5 x - 5)(7 - 4 x)+(- 2 x^{2} + 7 x - 8)(4 x + 5) " data-equation-content=" \displaystyle g'=(2 x^{2} + 5 x - 5)(7 - 4 x)+(- 2 x^{2} + 7 x - 8)(4 x + 5) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(x^{2} + 3 x + 9)(\left(7 - 4 x\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right))+(\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right))(2 x + 3)=\left(7 - 4 x\right) \left(x^{2} + 3 x + 9\right) \left(2 x^{2} + 5 x - 5\right) + \left(2 x + 3\right) \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right) \left(x^{2} + 3 x + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28x%5E%7B2%7D%20%2B%203%20x%20%2B%209%29%28%5Cleft%287%20-%204%20x%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%5Cright%29%20%2B%20%5Cleft%284%20x%20%2B%205%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%5Cright%29%29%2B%28%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%5Cright%29%29%282%20x%20%2B%203%29%3D%5Cleft%287%20-%204%20x%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%203%20x%20%2B%209%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%5Cright%29%20%2B%20%5Cleft%282%20x%20%2B%203%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%205%20x%20-%205%5Cright%29%20%2B%20%5Cleft%284%20x%20%2B%205%5Cright%29%20%5Cleft%28-%202%20x%5E%7B2%7D%20%2B%207%20x%20-%208%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%203%20x%20%2B%209%5Cright%29%20" alt="LaTeX: \displaystyle f'=(x^{2} + 3 x + 9)(\left(7 - 4 x\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right))+(\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right))(2 x + 3)=\left(7 - 4 x\right) \left(x^{2} + 3 x + 9\right) \left(2 x^{2} + 5 x - 5\right) + \left(2 x + 3\right) \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right) \left(x^{2} + 3 x + 9\right) " data-equation-content=" \displaystyle f'=(x^{2} + 3 x + 9)(\left(7 - 4 x\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right))+(\left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right))(2 x + 3)=\left(7 - 4 x\right) \left(x^{2} + 3 x + 9\right) \left(2 x^{2} + 5 x - 5\right) + \left(2 x + 3\right) \left(- 2 x^{2} + 7 x - 8\right) \left(2 x^{2} + 5 x - 5\right) + \left(4 x + 5\right) \left(- 2 x^{2} + 7 x - 8\right) \left(x^{2} + 3 x + 9\right) " /> </p> </p>