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Identifying \(\displaystyle f=3 x^{2} - 3 x + 6\) and \(\displaystyle g=\left(7 x^{2} + 5 x + 7\right) e^{x}\) and using the product rule with \(\displaystyle f=3 x^{2} - 3 x + 6 \implies f'=6 x - 3\). This leaves g as \(\displaystyle g = \left(7 x^{2} + 5 x + 7\right) e^{x}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=7 x^{2} + 5 x + 7 \implies f'=14 x + 5\) and \(\displaystyle g=e^{x} \implies g'=e^{x}\). Popping up a level gives \(\displaystyle g'=(e^{x})(14 x + 5)+(7 x^{2} + 5 x + 7)(e^{x})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(3 x^{2} - 3 x + 6)(\left(14 x + 5\right) e^{x} + \left(7 x^{2} + 5 x + 7\right) e^{x})+(\left(7 x^{2} + 5 x + 7\right) e^{x})(6 x - 3)=\left(6 x - 3\right) \left(7 x^{2} + 5 x + 7\right) e^{x} + \left(14 x + 5\right) \left(3 x^{2} - 3 x + 6\right) e^{x} + \left(3 x^{2} - 3 x + 6\right) \left(7 x^{2} + 5 x + 7\right) e^{x}\)

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\begin{question}Find the derivative of $y = (3 x^{2} - 3 x + 6)(7 x^{2} + 5 x + 7)(e^{x})$.
    \soln{9cm}{Identifying $f=3 x^{2} - 3 x + 6$ and $g=\left(7 x^{2} + 5 x + 7\right) e^{x}$ and using the product rule with $f=3 x^{2} - 3 x + 6 \implies f'=6 x - 3$. This leaves g as $g = \left(7 x^{2} + 5 x + 7\right) e^{x}$ which also requires the product rule. Pushing down in the new product rule $f=7 x^{2} + 5 x + 7 \implies f'=14 x + 5$ and $g=e^{x} \implies g'=e^{x}$. Popping up a level gives $g'=(e^{x})(14 x + 5)+(7 x^{2} + 5 x + 7)(e^{x})$Popping up again (Back to the original problem) gives $f'=(3 x^{2} - 3 x + 6)(\left(14 x + 5\right) e^{x} + \left(7 x^{2} + 5 x + 7\right) e^{x})+(\left(7 x^{2} + 5 x + 7\right) e^{x})(6 x - 3)=\left(6 x - 3\right) \left(7 x^{2} + 5 x + 7\right) e^{x} + \left(14 x + 5\right) \left(3 x^{2} - 3 x + 6\right) e^{x} + \left(3 x^{2} - 3 x + 6\right) \left(7 x^{2} + 5 x + 7\right) e^{x}$}

\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.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (3 x^{2} - 3 x + 6)(7 x^{2} + 5 x + 7)(e^{x}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%283%20x%5E%7B2%7D%20-%203%20x%20%2B%206%29%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%29%28e%5E%7Bx%7D%29%20" alt="LaTeX:  \displaystyle y = (3 x^{2} - 3 x + 6)(7 x^{2} + 5 x + 7)(e^{x}) " data-equation-content=" \displaystyle y = (3 x^{2} - 3 x + 6)(7 x^{2} + 5 x + 7)(e^{x}) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=3 x^{2} - 3 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20x%5E%7B2%7D%20-%203%20x%20%2B%206%20" alt="LaTeX:  \displaystyle f=3 x^{2} - 3 x + 6 " data-equation-content=" \displaystyle f=3 x^{2} - 3 x + 6 " />  and  <img class="equation_image" title=" \displaystyle g=\left(7 x^{2} + 5 x + 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g=\left(7 x^{2} + 5 x + 7\right) e^{x} " data-equation-content=" \displaystyle g=\left(7 x^{2} + 5 x + 7\right) e^{x} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=3 x^{2} - 3 x + 6 \implies f'=6 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20x%5E%7B2%7D%20-%203%20x%20%2B%206%20%5Cimplies%20f%27%3D6%20x%20-%203%20" alt="LaTeX:  \displaystyle f=3 x^{2} - 3 x + 6 \implies f'=6 x - 3 " data-equation-content=" \displaystyle f=3 x^{2} - 3 x + 6 \implies f'=6 x - 3 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(7 x^{2} + 5 x + 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g = \left(7 x^{2} + 5 x + 7\right) e^{x} " data-equation-content=" \displaystyle g = \left(7 x^{2} + 5 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=7 x^{2} + 5 x + 7 \implies f'=14 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%20%5Cimplies%20f%27%3D14%20x%20%2B%205%20" alt="LaTeX:  \displaystyle f=7 x^{2} + 5 x + 7 \implies f'=14 x + 5 " data-equation-content=" \displaystyle f=7 x^{2} + 5 x + 7 \implies f'=14 x + 5 " />  and  <img class="equation_image" title=" \displaystyle g=e^{x} \implies g'=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3De%5E%7Bx%7D%20%5Cimplies%20g%27%3De%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g=e^{x} \implies g'=e^{x} " data-equation-content=" \displaystyle g=e^{x} \implies g'=e^{x} " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(e^{x})(14 x + 5)+(7 x^{2} + 5 x + 7)(e^{x}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28e%5E%7Bx%7D%29%2814%20x%20%2B%205%29%2B%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%29%28e%5E%7Bx%7D%29%20" alt="LaTeX:  \displaystyle g'=(e^{x})(14 x + 5)+(7 x^{2} + 5 x + 7)(e^{x}) " data-equation-content=" \displaystyle g'=(e^{x})(14 x + 5)+(7 x^{2} + 5 x + 7)(e^{x}) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(3 x^{2} - 3 x + 6)(\left(14 x + 5\right) e^{x} + \left(7 x^{2} + 5 x + 7\right) e^{x})+(\left(7 x^{2} + 5 x + 7\right) e^{x})(6 x - 3)=\left(6 x - 3\right) \left(7 x^{2} + 5 x + 7\right) e^{x} + \left(14 x + 5\right) \left(3 x^{2} - 3 x + 6\right) e^{x} + \left(3 x^{2} - 3 x + 6\right) \left(7 x^{2} + 5 x + 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%283%20x%5E%7B2%7D%20-%203%20x%20%2B%206%29%28%5Cleft%2814%20x%20%2B%205%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%29%2B%28%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%29%286%20x%20-%203%29%3D%5Cleft%286%20x%20-%203%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%2814%20x%20%2B%205%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%283%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(3 x^{2} - 3 x + 6)(\left(14 x + 5\right) e^{x} + \left(7 x^{2} + 5 x + 7\right) e^{x})+(\left(7 x^{2} + 5 x + 7\right) e^{x})(6 x - 3)=\left(6 x - 3\right) \left(7 x^{2} + 5 x + 7\right) e^{x} + \left(14 x + 5\right) \left(3 x^{2} - 3 x + 6\right) e^{x} + \left(3 x^{2} - 3 x + 6\right) \left(7 x^{2} + 5 x + 7\right) e^{x} " data-equation-content=" \displaystyle f'=(3 x^{2} - 3 x + 6)(\left(14 x + 5\right) e^{x} + \left(7 x^{2} + 5 x + 7\right) e^{x})+(\left(7 x^{2} + 5 x + 7\right) e^{x})(6 x - 3)=\left(6 x - 3\right) \left(7 x^{2} + 5 x + 7\right) e^{x} + \left(14 x + 5\right) \left(3 x^{2} - 3 x + 6\right) e^{x} + \left(3 x^{2} - 3 x + 6\right) \left(7 x^{2} + 5 x + 7\right) e^{x} " /> </p> </p>