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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- x^{2} - x + 7)(9 x^{2} - 6 x + 7)(- 3 x^{2} - 6 x + 8)\).
Identifying \(\displaystyle f=- x^{2} - x + 7\) and \(\displaystyle g=\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right)\) and using the product rule with \(\displaystyle f=- x^{2} - x + 7 \implies f'=- 2 x - 1\). This leaves g as \(\displaystyle g = \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x^{2} - 6 x + 7 \implies f'=18 x - 6\) and \(\displaystyle g=- 3 x^{2} - 6 x + 8 \implies g'=- 6 x - 6\). Popping up a level gives \(\displaystyle g'=(- 3 x^{2} - 6 x + 8)(18 x - 6)+(9 x^{2} - 6 x + 7)(- 6 x - 6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- x^{2} - x + 7)(\left(- 6 x - 6\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right))+(\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right))(- 2 x - 1)=\left(- 6 x - 6\right) \left(- x^{2} - x + 7\right) \left(9 x^{2} - 6 x + 7\right) + \left(- 2 x - 1\right) \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right) \left(- x^{2} - x + 7\right)\)
\begin{question}Find the derivative of $y = (- x^{2} - x + 7)(9 x^{2} - 6 x + 7)(- 3 x^{2} - 6 x + 8)$.
\soln{9cm}{Identifying $f=- x^{2} - x + 7$ and $g=\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right)$ and using the product rule with $f=- x^{2} - x + 7 \implies f'=- 2 x - 1$. This leaves g as $g = \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x^{2} - 6 x + 7 \implies f'=18 x - 6$ and $g=- 3 x^{2} - 6 x + 8 \implies g'=- 6 x - 6$. Popping up a level gives $g'=(- 3 x^{2} - 6 x + 8)(18 x - 6)+(9 x^{2} - 6 x + 7)(- 6 x - 6)$Popping up again (Back to the original problem) gives $f'=(- x^{2} - x + 7)(\left(- 6 x - 6\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right))+(\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right))(- 2 x - 1)=\left(- 6 x - 6\right) \left(- x^{2} - x + 7\right) \left(9 x^{2} - 6 x + 7\right) + \left(- 2 x - 1\right) \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right) \left(- x^{2} - x + 7\right)$}
\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 = (- x^{2} - x + 7)(9 x^{2} - 6 x + 7)(- 3 x^{2} - 6 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%20x%5E%7B2%7D%20-%20x%20%2B%207%29%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%29%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle y = (- x^{2} - x + 7)(9 x^{2} - 6 x + 7)(- 3 x^{2} - 6 x + 8) " data-equation-content=" \displaystyle y = (- x^{2} - x + 7)(9 x^{2} - 6 x + 7)(- 3 x^{2} - 6 x + 8) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- x^{2} - x + 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%20x%5E%7B2%7D%20-%20x%20%2B%207%20" alt="LaTeX: \displaystyle f=- x^{2} - x + 7 " data-equation-content=" \displaystyle f=- x^{2} - x + 7 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) " data-equation-content=" \displaystyle g=\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- x^{2} - x + 7 \implies f'=- 2 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%20x%5E%7B2%7D%20-%20x%20%2B%207%20%5Cimplies%20f%27%3D-%202%20x%20-%201%20" alt="LaTeX: \displaystyle f=- x^{2} - x + 7 \implies f'=- 2 x - 1 " data-equation-content=" \displaystyle f=- x^{2} - x + 7 \implies f'=- 2 x - 1 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) " data-equation-content=" \displaystyle g = \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=9 x^{2} - 6 x + 7 \implies f'=18 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20-%206%20x%20%2B%207%20%5Cimplies%20f%27%3D18%20x%20-%206%20" alt="LaTeX: \displaystyle f=9 x^{2} - 6 x + 7 \implies f'=18 x - 6 " data-equation-content=" \displaystyle f=9 x^{2} - 6 x + 7 \implies f'=18 x - 6 " /> and <img class="equation_image" title=" \displaystyle g=- 3 x^{2} - 6 x + 8 \implies g'=- 6 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%20%5Cimplies%20g%27%3D-%206%20x%20-%206%20" alt="LaTeX: \displaystyle g=- 3 x^{2} - 6 x + 8 \implies g'=- 6 x - 6 " data-equation-content=" \displaystyle g=- 3 x^{2} - 6 x + 8 \implies g'=- 6 x - 6 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 3 x^{2} - 6 x + 8)(18 x - 6)+(9 x^{2} - 6 x + 7)(- 6 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%29%2818%20x%20-%206%29%2B%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%29%28-%206%20x%20-%206%29%20" alt="LaTeX: \displaystyle g'=(- 3 x^{2} - 6 x + 8)(18 x - 6)+(9 x^{2} - 6 x + 7)(- 6 x - 6) " data-equation-content=" \displaystyle g'=(- 3 x^{2} - 6 x + 8)(18 x - 6)+(9 x^{2} - 6 x + 7)(- 6 x - 6) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- x^{2} - x + 7)(\left(- 6 x - 6\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right))+(\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right))(- 2 x - 1)=\left(- 6 x - 6\right) \left(- x^{2} - x + 7\right) \left(9 x^{2} - 6 x + 7\right) + \left(- 2 x - 1\right) \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right) \left(- x^{2} - x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%20x%5E%7B2%7D%20-%20x%20%2B%207%29%28%5Cleft%28-%206%20x%20-%206%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%5Cright%29%20%2B%20%5Cleft%2818%20x%20-%206%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%5Cright%29%29%2B%28%5Cleft%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%5Cright%29%29%28-%202%20x%20-%201%29%3D%5Cleft%28-%206%20x%20-%206%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20-%20x%20%2B%207%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%5Cright%29%20%2B%20%5Cleft%28-%202%20x%20-%201%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%206%20x%20%2B%207%5Cright%29%20%2B%20%5Cleft%2818%20x%20-%206%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20-%206%20x%20%2B%208%5Cright%29%20%5Cleft%28-%20x%5E%7B2%7D%20-%20x%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- x^{2} - x + 7)(\left(- 6 x - 6\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right))+(\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right))(- 2 x - 1)=\left(- 6 x - 6\right) \left(- x^{2} - x + 7\right) \left(9 x^{2} - 6 x + 7\right) + \left(- 2 x - 1\right) \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right) \left(- x^{2} - x + 7\right) " data-equation-content=" \displaystyle f'=(- x^{2} - x + 7)(\left(- 6 x - 6\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right))+(\left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right))(- 2 x - 1)=\left(- 6 x - 6\right) \left(- x^{2} - x + 7\right) \left(9 x^{2} - 6 x + 7\right) + \left(- 2 x - 1\right) \left(- 3 x^{2} - 6 x + 8\right) \left(9 x^{2} - 6 x + 7\right) + \left(18 x - 6\right) \left(- 3 x^{2} - 6 x + 8\right) \left(- x^{2} - x + 7\right) " /> </p> </p>