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Identifying \(\displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7\) and \(\displaystyle g=\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x}\) and using the product rule with \(\displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 \implies f'=- 18 x^{2} - 16 x + 9\). This leaves g as \(\displaystyle g = \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=e^{x} \implies f'=e^{x}\) and \(\displaystyle g=6 x^{3} - 4 x^{2} + x + 3 \implies g'=18 x^{2} - 8 x + 1\). Popping up a level gives \(\displaystyle g'=(6 x^{3} - 4 x^{2} + x + 3)(e^{x})+(e^{x})(18 x^{2} - 8 x + 1)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 6 x^{3} - 8 x^{2} + 9 x - 7)(\left(18 x^{2} - 8 x + 1\right) e^{x} + \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})+(\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})(- 18 x^{2} - 16 x + 9)=\left(- 18 x^{2} - 16 x + 9\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} + \left(18 x^{2} - 8 x + 1\right) \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) e^{x} + \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x}\)

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\begin{question}Find the derivative of $y = (- 6 x^{3} - 8 x^{2} + 9 x - 7)(e^{x})(6 x^{3} - 4 x^{2} + x + 3)$.
    \soln{9cm}{Identifying $f=- 6 x^{3} - 8 x^{2} + 9 x - 7$ and $g=\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x}$ and using the product rule with $f=- 6 x^{3} - 8 x^{2} + 9 x - 7 \implies f'=- 18 x^{2} - 16 x + 9$. This leaves g as $g = \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x}$ which also requires the product rule. Pushing down in the new product rule $f=e^{x} \implies f'=e^{x}$ and $g=6 x^{3} - 4 x^{2} + x + 3 \implies g'=18 x^{2} - 8 x + 1$. Popping up a level gives $g'=(6 x^{3} - 4 x^{2} + x + 3)(e^{x})+(e^{x})(18 x^{2} - 8 x + 1)$Popping up again (Back to the original problem) gives $f'=(- 6 x^{3} - 8 x^{2} + 9 x - 7)(\left(18 x^{2} - 8 x + 1\right) e^{x} + \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})+(\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})(- 18 x^{2} - 16 x + 9)=\left(- 18 x^{2} - 16 x + 9\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} + \left(18 x^{2} - 8 x + 1\right) \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) e^{x} + \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x}$}

\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}

HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (- 6 x^{3} - 8 x^{2} + 9 x - 7)(e^{x})(6 x^{3} - 4 x^{2} + x + 3) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%206%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20%2B%209%20x%20-%207%29%28e%5E%7Bx%7D%29%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%29%20" alt="LaTeX:  \displaystyle y = (- 6 x^{3} - 8 x^{2} + 9 x - 7)(e^{x})(6 x^{3} - 4 x^{2} + x + 3) " data-equation-content=" \displaystyle y = (- 6 x^{3} - 8 x^{2} + 9 x - 7)(e^{x})(6 x^{3} - 4 x^{2} + x + 3) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20%2B%209%20x%20-%207%20" alt="LaTeX:  \displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 " data-equation-content=" \displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 " />  and  <img class="equation_image" title=" \displaystyle g=\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g=\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " data-equation-content=" \displaystyle g=\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 \implies f'=- 18 x^{2} - 16 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20%2B%209%20x%20-%207%20%5Cimplies%20f%27%3D-%2018%20x%5E%7B2%7D%20-%2016%20x%20%2B%209%20" alt="LaTeX:  \displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 \implies f'=- 18 x^{2} - 16 x + 9 " data-equation-content=" \displaystyle f=- 6 x^{3} - 8 x^{2} + 9 x - 7 \implies f'=- 18 x^{2} - 16 x + 9 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g = \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " data-equation-content=" \displaystyle g = \left(6 x^{3} - 4 x^{2} + x + 3\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=6 x^{3} - 4 x^{2} + x + 3 \implies g'=18 x^{2} - 8 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20g%3D6%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%20%5Cimplies%20g%27%3D18%20x%5E%7B2%7D%20-%208%20x%20%2B%201%20" alt="LaTeX:  \displaystyle g=6 x^{3} - 4 x^{2} + x + 3 \implies g'=18 x^{2} - 8 x + 1 " data-equation-content=" \displaystyle g=6 x^{3} - 4 x^{2} + x + 3 \implies g'=18 x^{2} - 8 x + 1 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(6 x^{3} - 4 x^{2} + x + 3)(e^{x})+(e^{x})(18 x^{2} - 8 x + 1) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%29%28e%5E%7Bx%7D%29%2B%28e%5E%7Bx%7D%29%2818%20x%5E%7B2%7D%20-%208%20x%20%2B%201%29%20" alt="LaTeX:  \displaystyle g'=(6 x^{3} - 4 x^{2} + x + 3)(e^{x})+(e^{x})(18 x^{2} - 8 x + 1) " data-equation-content=" \displaystyle g'=(6 x^{3} - 4 x^{2} + x + 3)(e^{x})+(e^{x})(18 x^{2} - 8 x + 1) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 6 x^{3} - 8 x^{2} + 9 x - 7)(\left(18 x^{2} - 8 x + 1\right) e^{x} + \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})+(\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})(- 18 x^{2} - 16 x + 9)=\left(- 18 x^{2} - 16 x + 9\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} + \left(18 x^{2} - 8 x + 1\right) \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) e^{x} + \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%206%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20%2B%209%20x%20-%207%29%28%5Cleft%2818%20x%5E%7B2%7D%20-%208%20x%20%2B%201%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%29%2B%28%5Cleft%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%29%28-%2018%20x%5E%7B2%7D%20-%2016%20x%20%2B%209%29%3D%5Cleft%28-%2018%20x%5E%7B2%7D%20-%2016%20x%20%2B%209%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%2818%20x%5E%7B2%7D%20-%208%20x%20%2B%201%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%20x%20%2B%203%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(- 6 x^{3} - 8 x^{2} + 9 x - 7)(\left(18 x^{2} - 8 x + 1\right) e^{x} + \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})+(\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})(- 18 x^{2} - 16 x + 9)=\left(- 18 x^{2} - 16 x + 9\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} + \left(18 x^{2} - 8 x + 1\right) \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) e^{x} + \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " data-equation-content=" \displaystyle f'=(- 6 x^{3} - 8 x^{2} + 9 x - 7)(\left(18 x^{2} - 8 x + 1\right) e^{x} + \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})+(\left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x})(- 18 x^{2} - 16 x + 9)=\left(- 18 x^{2} - 16 x + 9\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} + \left(18 x^{2} - 8 x + 1\right) \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) e^{x} + \left(- 6 x^{3} - 8 x^{2} + 9 x - 7\right) \left(6 x^{3} - 4 x^{2} + x + 3\right) e^{x} " /> </p> </p>