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

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

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

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HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (9 x^{2} + 7 x - 9)(e^{x})(2 x^{2} + 4 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%289%20x%5E%7B2%7D%20%2B%207%20x%20-%209%29%28e%5E%7Bx%7D%29%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%29%20" alt="LaTeX:  \displaystyle y = (9 x^{2} + 7 x - 9)(e^{x})(2 x^{2} + 4 x - 6) " data-equation-content=" \displaystyle y = (9 x^{2} + 7 x - 9)(e^{x})(2 x^{2} + 4 x - 6) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=9 x^{2} + 7 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20%2B%207%20x%20-%209%20" alt="LaTeX:  \displaystyle f=9 x^{2} + 7 x - 9 " data-equation-content=" \displaystyle f=9 x^{2} + 7 x - 9 " />  and  <img class="equation_image" title=" \displaystyle g=\left(2 x^{2} + 4 x - 6\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g=\left(2 x^{2} + 4 x - 6\right) e^{x} " data-equation-content=" \displaystyle g=\left(2 x^{2} + 4 x - 6\right) e^{x} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=9 x^{2} + 7 x - 9 \implies f'=18 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20%2B%207%20x%20-%209%20%5Cimplies%20f%27%3D18%20x%20%2B%207%20" alt="LaTeX:  \displaystyle f=9 x^{2} + 7 x - 9 \implies f'=18 x + 7 " data-equation-content=" \displaystyle f=9 x^{2} + 7 x - 9 \implies f'=18 x + 7 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(2 x^{2} + 4 x - 6\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g = \left(2 x^{2} + 4 x - 6\right) e^{x} " data-equation-content=" \displaystyle g = \left(2 x^{2} + 4 x - 6\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=2 x^{2} + 4 x - 6 \implies g'=4 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20g%3D2%20x%5E%7B2%7D%20%2B%204%20x%20-%206%20%5Cimplies%20g%27%3D4%20x%20%2B%204%20" alt="LaTeX:  \displaystyle g=2 x^{2} + 4 x - 6 \implies g'=4 x + 4 " data-equation-content=" \displaystyle g=2 x^{2} + 4 x - 6 \implies g'=4 x + 4 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(2 x^{2} + 4 x - 6)(e^{x})+(e^{x})(4 x + 4) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%29%28e%5E%7Bx%7D%29%2B%28e%5E%7Bx%7D%29%284%20x%20%2B%204%29%20" alt="LaTeX:  \displaystyle g'=(2 x^{2} + 4 x - 6)(e^{x})+(e^{x})(4 x + 4) " data-equation-content=" \displaystyle g'=(2 x^{2} + 4 x - 6)(e^{x})+(e^{x})(4 x + 4) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(9 x^{2} + 7 x - 9)(\left(4 x + 4\right) e^{x} + \left(2 x^{2} + 4 x - 6\right) e^{x})+(\left(2 x^{2} + 4 x - 6\right) e^{x})(18 x + 7)=\left(4 x + 4\right) \left(9 x^{2} + 7 x - 9\right) e^{x} + \left(18 x + 7\right) \left(2 x^{2} + 4 x - 6\right) e^{x} + \left(2 x^{2} + 4 x - 6\right) \left(9 x^{2} + 7 x - 9\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%289%20x%5E%7B2%7D%20%2B%207%20x%20-%209%29%28%5Cleft%284%20x%20%2B%204%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%5Cright%29%20e%5E%7Bx%7D%29%2B%28%5Cleft%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%5Cright%29%20e%5E%7Bx%7D%29%2818%20x%20%2B%207%29%3D%5Cleft%284%20x%20%2B%204%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%207%20x%20-%209%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%2818%20x%20%2B%207%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%282%20x%5E%7B2%7D%20%2B%204%20x%20-%206%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%207%20x%20-%209%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(9 x^{2} + 7 x - 9)(\left(4 x + 4\right) e^{x} + \left(2 x^{2} + 4 x - 6\right) e^{x})+(\left(2 x^{2} + 4 x - 6\right) e^{x})(18 x + 7)=\left(4 x + 4\right) \left(9 x^{2} + 7 x - 9\right) e^{x} + \left(18 x + 7\right) \left(2 x^{2} + 4 x - 6\right) e^{x} + \left(2 x^{2} + 4 x - 6\right) \left(9 x^{2} + 7 x - 9\right) e^{x} " data-equation-content=" \displaystyle f'=(9 x^{2} + 7 x - 9)(\left(4 x + 4\right) e^{x} + \left(2 x^{2} + 4 x - 6\right) e^{x})+(\left(2 x^{2} + 4 x - 6\right) e^{x})(18 x + 7)=\left(4 x + 4\right) \left(9 x^{2} + 7 x - 9\right) e^{x} + \left(18 x + 7\right) \left(2 x^{2} + 4 x - 6\right) e^{x} + \left(2 x^{2} + 4 x - 6\right) \left(9 x^{2} + 7 x - 9\right) e^{x} " /> </p> </p>