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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 6 x^{2} - 4 x + 6)(- 3 x^{2} - 7 x - 4)(5 x^{2} + 9 x + 8)\).
Identifying \(\displaystyle f=- 6 x^{2} - 4 x + 6\) and \(\displaystyle g=\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right)\) and using the product rule with \(\displaystyle f=- 6 x^{2} - 4 x + 6 \implies f'=- 12 x - 4\). This leaves g as \(\displaystyle g = \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 3 x^{2} - 7 x - 4 \implies f'=- 6 x - 7\) and \(\displaystyle g=5 x^{2} + 9 x + 8 \implies g'=10 x + 9\). Popping up a level gives \(\displaystyle g'=(5 x^{2} + 9 x + 8)(- 6 x - 7)+(- 3 x^{2} - 7 x - 4)(10 x + 9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 6 x^{2} - 4 x + 6)(\left(- 6 x - 7\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 3 x^{2} - 7 x - 4\right))+(\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right))(- 12 x - 4)=\left(- 12 x - 4\right) \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) + \left(- 6 x - 7\right) \left(- 6 x^{2} - 4 x + 6\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 6 x^{2} - 4 x + 6\right) \left(- 3 x^{2} - 7 x - 4\right)\)
\begin{question}Find the derivative of $y = (- 6 x^{2} - 4 x + 6)(- 3 x^{2} - 7 x - 4)(5 x^{2} + 9 x + 8)$.
\soln{9cm}{Identifying $f=- 6 x^{2} - 4 x + 6$ and $g=\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right)$ and using the product rule with $f=- 6 x^{2} - 4 x + 6 \implies f'=- 12 x - 4$. This leaves g as $g = \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 3 x^{2} - 7 x - 4 \implies f'=- 6 x - 7$ and $g=5 x^{2} + 9 x + 8 \implies g'=10 x + 9$. Popping up a level gives $g'=(5 x^{2} + 9 x + 8)(- 6 x - 7)+(- 3 x^{2} - 7 x - 4)(10 x + 9)$Popping up again (Back to the original problem) gives $f'=(- 6 x^{2} - 4 x + 6)(\left(- 6 x - 7\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 3 x^{2} - 7 x - 4\right))+(\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right))(- 12 x - 4)=\left(- 12 x - 4\right) \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) + \left(- 6 x - 7\right) \left(- 6 x^{2} - 4 x + 6\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 6 x^{2} - 4 x + 6\right) \left(- 3 x^{2} - 7 x - 4\right)$}
\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 = (- 6 x^{2} - 4 x + 6)(- 3 x^{2} - 7 x - 4)(5 x^{2} + 9 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%206%20x%5E%7B2%7D%20-%204%20x%20%2B%206%29%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%29%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle y = (- 6 x^{2} - 4 x + 6)(- 3 x^{2} - 7 x - 4)(5 x^{2} + 9 x + 8) " data-equation-content=" \displaystyle y = (- 6 x^{2} - 4 x + 6)(- 3 x^{2} - 7 x - 4)(5 x^{2} + 9 x + 8) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 6 x^{2} - 4 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B2%7D%20-%204%20x%20%2B%206%20" alt="LaTeX: \displaystyle f=- 6 x^{2} - 4 x + 6 " data-equation-content=" \displaystyle f=- 6 x^{2} - 4 x + 6 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) " data-equation-content=" \displaystyle g=\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 6 x^{2} - 4 x + 6 \implies f'=- 12 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B2%7D%20-%204%20x%20%2B%206%20%5Cimplies%20f%27%3D-%2012%20x%20-%204%20" alt="LaTeX: \displaystyle f=- 6 x^{2} - 4 x + 6 \implies f'=- 12 x - 4 " data-equation-content=" \displaystyle f=- 6 x^{2} - 4 x + 6 \implies f'=- 12 x - 4 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) " data-equation-content=" \displaystyle g = \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 3 x^{2} - 7 x - 4 \implies f'=- 6 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%20%5Cimplies%20f%27%3D-%206%20x%20-%207%20" alt="LaTeX: \displaystyle f=- 3 x^{2} - 7 x - 4 \implies f'=- 6 x - 7 " data-equation-content=" \displaystyle f=- 3 x^{2} - 7 x - 4 \implies f'=- 6 x - 7 " /> and <img class="equation_image" title=" \displaystyle g=5 x^{2} + 9 x + 8 \implies g'=10 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D5%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%20%5Cimplies%20g%27%3D10%20x%20%2B%209%20" alt="LaTeX: \displaystyle g=5 x^{2} + 9 x + 8 \implies g'=10 x + 9 " data-equation-content=" \displaystyle g=5 x^{2} + 9 x + 8 \implies g'=10 x + 9 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(5 x^{2} + 9 x + 8)(- 6 x - 7)+(- 3 x^{2} - 7 x - 4)(10 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%29%28-%206%20x%20-%207%29%2B%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%29%2810%20x%20%2B%209%29%20" alt="LaTeX: \displaystyle g'=(5 x^{2} + 9 x + 8)(- 6 x - 7)+(- 3 x^{2} - 7 x - 4)(10 x + 9) " data-equation-content=" \displaystyle g'=(5 x^{2} + 9 x + 8)(- 6 x - 7)+(- 3 x^{2} - 7 x - 4)(10 x + 9) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 6 x^{2} - 4 x + 6)(\left(- 6 x - 7\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 3 x^{2} - 7 x - 4\right))+(\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right))(- 12 x - 4)=\left(- 12 x - 4\right) \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) + \left(- 6 x - 7\right) \left(- 6 x^{2} - 4 x + 6\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 6 x^{2} - 4 x + 6\right) \left(- 3 x^{2} - 7 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%206%20x%5E%7B2%7D%20-%204%20x%20%2B%206%29%28%5Cleft%28-%206%20x%20-%207%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%2810%20x%20%2B%209%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%5Cright%29%29%2B%28%5Cleft%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%5Cright%29%29%28-%2012%20x%20-%204%29%3D%5Cleft%28-%2012%20x%20-%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%28-%206%20x%20-%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20-%204%20x%20%2B%206%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%209%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%2810%20x%20%2B%209%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20-%204%20x%20%2B%206%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%207%20x%20-%204%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 6 x^{2} - 4 x + 6)(\left(- 6 x - 7\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 3 x^{2} - 7 x - 4\right))+(\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right))(- 12 x - 4)=\left(- 12 x - 4\right) \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) + \left(- 6 x - 7\right) \left(- 6 x^{2} - 4 x + 6\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 6 x^{2} - 4 x + 6\right) \left(- 3 x^{2} - 7 x - 4\right) " data-equation-content=" \displaystyle f'=(- 6 x^{2} - 4 x + 6)(\left(- 6 x - 7\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 3 x^{2} - 7 x - 4\right))+(\left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right))(- 12 x - 4)=\left(- 12 x - 4\right) \left(- 3 x^{2} - 7 x - 4\right) \left(5 x^{2} + 9 x + 8\right) + \left(- 6 x - 7\right) \left(- 6 x^{2} - 4 x + 6\right) \left(5 x^{2} + 9 x + 8\right) + \left(10 x + 9\right) \left(- 6 x^{2} - 4 x + 6\right) \left(- 3 x^{2} - 7 x - 4\right) " /> </p> </p>