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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (\cos{\left(x \right)})(6 x^{2} + 6 x + 8)(- 9 x^{2} - 7 x + 1)\).
Identifying \(\displaystyle f=\cos{\left(x \right)}\) and \(\displaystyle g=\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right)\) and using the product rule with \(\displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)}\). This leaves g as \(\displaystyle g = \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=6 x^{2} + 6 x + 8 \implies f'=12 x + 6\) and \(\displaystyle g=- 9 x^{2} - 7 x + 1 \implies g'=- 18 x - 7\). Popping up a level gives \(\displaystyle g'=(- 9 x^{2} - 7 x + 1)(12 x + 6)+(6 x^{2} + 6 x + 8)(- 18 x - 7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(\cos{\left(x \right)})(\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right))+(\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right))(- \sin{\left(x \right)})=\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) \cos{\left(x \right)} + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right) \cos{\left(x \right)} - \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) \sin{\left(x \right)}\)
\begin{question}Find the derivative of $y = (\cos{\left(x \right)})(6 x^{2} + 6 x + 8)(- 9 x^{2} - 7 x + 1)$.
\soln{9cm}{Identifying $f=\cos{\left(x \right)}$ and $g=\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right)$ and using the product rule with $f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)}$. This leaves g as $g = \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right)$ which also requires the product rule. Pushing down in the new product rule $f=6 x^{2} + 6 x + 8 \implies f'=12 x + 6$ and $g=- 9 x^{2} - 7 x + 1 \implies g'=- 18 x - 7$. Popping up a level gives $g'=(- 9 x^{2} - 7 x + 1)(12 x + 6)+(6 x^{2} + 6 x + 8)(- 18 x - 7)$Popping up again (Back to the original problem) gives $f'=(\cos{\left(x \right)})(\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right))+(\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right))(- \sin{\left(x \right)})=\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) \cos{\left(x \right)} + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right) \cos{\left(x \right)} - \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) \sin{\left(x \right)}$}
\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 = (\cos{\left(x \right)})(6 x^{2} + 6 x + 8)(- 9 x^{2} - 7 x + 1) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%29%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%29%20" alt="LaTeX: \displaystyle y = (\cos{\left(x \right)})(6 x^{2} + 6 x + 8)(- 9 x^{2} - 7 x + 1) " data-equation-content=" \displaystyle y = (\cos{\left(x \right)})(6 x^{2} + 6 x + 8)(- 9 x^{2} - 7 x + 1) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f=\cos{\left(x \right)} " data-equation-content=" \displaystyle f=\cos{\left(x \right)} " /> and <img class="equation_image" title=" \displaystyle g=\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) " data-equation-content=" \displaystyle g=\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)} " data-equation-content=" \displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)} " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) " data-equation-content=" \displaystyle g = \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=6 x^{2} + 6 x + 8 \implies f'=12 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D6%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%20%5Cimplies%20f%27%3D12%20x%20%2B%206%20" alt="LaTeX: \displaystyle f=6 x^{2} + 6 x + 8 \implies f'=12 x + 6 " data-equation-content=" \displaystyle f=6 x^{2} + 6 x + 8 \implies f'=12 x + 6 " /> and <img class="equation_image" title=" \displaystyle g=- 9 x^{2} - 7 x + 1 \implies g'=- 18 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%20%5Cimplies%20g%27%3D-%2018%20x%20-%207%20" alt="LaTeX: \displaystyle g=- 9 x^{2} - 7 x + 1 \implies g'=- 18 x - 7 " data-equation-content=" \displaystyle g=- 9 x^{2} - 7 x + 1 \implies g'=- 18 x - 7 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 9 x^{2} - 7 x + 1)(12 x + 6)+(6 x^{2} + 6 x + 8)(- 18 x - 7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%29%2812%20x%20%2B%206%29%2B%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%29%28-%2018%20x%20-%207%29%20" alt="LaTeX: \displaystyle g'=(- 9 x^{2} - 7 x + 1)(12 x + 6)+(6 x^{2} + 6 x + 8)(- 18 x - 7) " data-equation-content=" \displaystyle g'=(- 9 x^{2} - 7 x + 1)(12 x + 6)+(6 x^{2} + 6 x + 8)(- 18 x - 7) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(\cos{\left(x \right)})(\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right))+(\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right))(- \sin{\left(x \right)})=\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) \cos{\left(x \right)} + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right) \cos{\left(x \right)} - \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Cleft%28-%2018%20x%20-%207%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%2812%20x%20%2B%206%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%5Cright%29%29%2B%28%5Cleft%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%3D%5Cleft%28-%2018%20x%20-%207%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2812%20x%20%2B%206%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%207%20x%20%2B%201%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f'=(\cos{\left(x \right)})(\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right))+(\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right))(- \sin{\left(x \right)})=\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) \cos{\left(x \right)} + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right) \cos{\left(x \right)} - \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle f'=(\cos{\left(x \right)})(\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right))+(\left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right))(- \sin{\left(x \right)})=\left(- 18 x - 7\right) \left(6 x^{2} + 6 x + 8\right) \cos{\left(x \right)} + \left(12 x + 6\right) \left(- 9 x^{2} - 7 x + 1\right) \cos{\left(x \right)} - \left(- 9 x^{2} - 7 x + 1\right) \left(6 x^{2} + 6 x + 8\right) \sin{\left(x \right)} " /> </p> </p>