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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (8 x^{3} - 8 x^{2} - 6 x - 9)(e^{x})(- x^{3} + 7 x^{2} + 6 x + 7)\).
Identifying \(\displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9\) and \(\displaystyle g=\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x}\) and using the product rule with \(\displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 \implies f'=24 x^{2} - 16 x - 6\). This leaves g as \(\displaystyle g = \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=e^{x} \implies f'=e^{x}\) and \(\displaystyle g=- x^{3} + 7 x^{2} + 6 x + 7 \implies g'=- 3 x^{2} + 14 x + 6\). Popping up a level gives \(\displaystyle g'=(- x^{3} + 7 x^{2} + 6 x + 7)(e^{x})+(e^{x})(- 3 x^{2} + 14 x + 6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(8 x^{3} - 8 x^{2} - 6 x - 9)(\left(- 3 x^{2} + 14 x + 6\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})+(\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})(24 x^{2} - 16 x - 6)=\left(- 3 x^{2} + 14 x + 6\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} + \left(24 x^{2} - 16 x - 6\right) \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x}\)
\begin{question}Find the derivative of $y = (8 x^{3} - 8 x^{2} - 6 x - 9)(e^{x})(- x^{3} + 7 x^{2} + 6 x + 7)$.
\soln{9cm}{Identifying $f=8 x^{3} - 8 x^{2} - 6 x - 9$ and $g=\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x}$ and using the product rule with $f=8 x^{3} - 8 x^{2} - 6 x - 9 \implies f'=24 x^{2} - 16 x - 6$. This leaves g as $g = \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x}$ which also requires the product rule. Pushing down in the new product rule $f=e^{x} \implies f'=e^{x}$ and $g=- x^{3} + 7 x^{2} + 6 x + 7 \implies g'=- 3 x^{2} + 14 x + 6$. Popping up a level gives $g'=(- x^{3} + 7 x^{2} + 6 x + 7)(e^{x})+(e^{x})(- 3 x^{2} + 14 x + 6)$Popping up again (Back to the original problem) gives $f'=(8 x^{3} - 8 x^{2} - 6 x - 9)(\left(- 3 x^{2} + 14 x + 6\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})+(\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})(24 x^{2} - 16 x - 6)=\left(- 3 x^{2} + 14 x + 6\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} + \left(24 x^{2} - 16 x - 6\right) \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x}$}
\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^{3} - 8 x^{2} - 6 x - 9)(e^{x})(- x^{3} + 7 x^{2} + 6 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%288%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%206%20x%20-%209%29%28e%5E%7Bx%7D%29%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%29%20" alt="LaTeX: \displaystyle y = (8 x^{3} - 8 x^{2} - 6 x - 9)(e^{x})(- x^{3} + 7 x^{2} + 6 x + 7) " data-equation-content=" \displaystyle y = (8 x^{3} - 8 x^{2} - 6 x - 9)(e^{x})(- x^{3} + 7 x^{2} + 6 x + 7) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%206%20x%20-%209%20" alt="LaTeX: \displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 " data-equation-content=" \displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 " /> and <img class="equation_image" title=" \displaystyle g=\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle g=\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} " data-equation-content=" \displaystyle g=\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 \implies f'=24 x^{2} - 16 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%206%20x%20-%209%20%5Cimplies%20f%27%3D24%20x%5E%7B2%7D%20-%2016%20x%20-%206%20" alt="LaTeX: \displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 \implies f'=24 x^{2} - 16 x - 6 " data-equation-content=" \displaystyle f=8 x^{3} - 8 x^{2} - 6 x - 9 \implies f'=24 x^{2} - 16 x - 6 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle g = \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} " data-equation-content=" \displaystyle g = \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=e^{x} \implies f'=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20%5Cimplies%20f%27%3De%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f=e^{x} \implies f'=e^{x} " data-equation-content=" \displaystyle f=e^{x} \implies f'=e^{x} " /> and <img class="equation_image" title=" \displaystyle g=- x^{3} + 7 x^{2} + 6 x + 7 \implies g'=- 3 x^{2} + 14 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%20%5Cimplies%20g%27%3D-%203%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%20" alt="LaTeX: \displaystyle g=- x^{3} + 7 x^{2} + 6 x + 7 \implies g'=- 3 x^{2} + 14 x + 6 " data-equation-content=" \displaystyle g=- x^{3} + 7 x^{2} + 6 x + 7 \implies g'=- 3 x^{2} + 14 x + 6 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- x^{3} + 7 x^{2} + 6 x + 7)(e^{x})+(e^{x})(- 3 x^{2} + 14 x + 6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%29%28e%5E%7Bx%7D%29%2B%28e%5E%7Bx%7D%29%28-%203%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%29%20" alt="LaTeX: \displaystyle g'=(- x^{3} + 7 x^{2} + 6 x + 7)(e^{x})+(e^{x})(- 3 x^{2} + 14 x + 6) " data-equation-content=" \displaystyle g'=(- x^{3} + 7 x^{2} + 6 x + 7)(e^{x})+(e^{x})(- 3 x^{2} + 14 x + 6) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(8 x^{3} - 8 x^{2} - 6 x - 9)(\left(- 3 x^{2} + 14 x + 6\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})+(\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})(24 x^{2} - 16 x - 6)=\left(- 3 x^{2} + 14 x + 6\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} + \left(24 x^{2} - 16 x - 6\right) \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%288%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%206%20x%20-%209%29%28%5Cleft%28-%203%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%29%2B%28%5Cleft%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%29%2824%20x%5E%7B2%7D%20-%2016%20x%20-%206%29%3D%5Cleft%28-%203%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%5Cright%29%20%5Cleft%288%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%2824%20x%5E%7B2%7D%20-%2016%20x%20-%206%5Cright%29%20%5Cleft%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20%2B%207%5Cright%29%20%5Cleft%288%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%206%20x%20-%209%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f'=(8 x^{3} - 8 x^{2} - 6 x - 9)(\left(- 3 x^{2} + 14 x + 6\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})+(\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})(24 x^{2} - 16 x - 6)=\left(- 3 x^{2} + 14 x + 6\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} + \left(24 x^{2} - 16 x - 6\right) \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} " data-equation-content=" \displaystyle f'=(8 x^{3} - 8 x^{2} - 6 x - 9)(\left(- 3 x^{2} + 14 x + 6\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})+(\left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x})(24 x^{2} - 16 x - 6)=\left(- 3 x^{2} + 14 x + 6\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} + \left(24 x^{2} - 16 x - 6\right) \left(- x^{3} + 7 x^{2} + 6 x + 7\right) e^{x} + \left(- x^{3} + 7 x^{2} + 6 x + 7\right) \left(8 x^{3} - 8 x^{2} - 6 x - 9\right) e^{x} " /> </p> </p>