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Identifying \(\displaystyle f=7 x^{2} + 5 x + 4\) and \(\displaystyle g=\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)}\) and using the product rule with \(\displaystyle f=7 x^{2} + 5 x + 4 \implies f'=14 x + 5\). This leaves g as \(\displaystyle g = \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}\) and \(\displaystyle g=- 8 x^{2} - 8 x - 3 \implies g'=- 16 x - 8\). Popping up a level gives \(\displaystyle g'=(- 8 x^{2} - 8 x - 3)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 16 x - 8)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(7 x^{2} + 5 x + 4)(\left(- 16 x - 8\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \cos{\left(x \right)})+(\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)})(14 x + 5)=\left(- 16 x - 8\right) \left(7 x^{2} + 5 x + 4\right) \sin{\left(x \right)} + \left(14 x + 5\right) \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \left(7 x^{2} + 5 x + 4\right) \cos{\left(x \right)}\)

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\begin{question}Find the derivative of $y = (7 x^{2} + 5 x + 4)(\sin{\left(x \right)})(- 8 x^{2} - 8 x - 3)$.
    \soln{9cm}{Identifying $f=7 x^{2} + 5 x + 4$ and $g=\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)}$ and using the product rule with $f=7 x^{2} + 5 x + 4 \implies f'=14 x + 5$. This leaves g as $g = \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}$ and $g=- 8 x^{2} - 8 x - 3 \implies g'=- 16 x - 8$. Popping up a level gives $g'=(- 8 x^{2} - 8 x - 3)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 16 x - 8)$Popping up again (Back to the original problem) gives $f'=(7 x^{2} + 5 x + 4)(\left(- 16 x - 8\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \cos{\left(x \right)})+(\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)})(14 x + 5)=\left(- 16 x - 8\right) \left(7 x^{2} + 5 x + 4\right) \sin{\left(x \right)} + \left(14 x + 5\right) \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \left(7 x^{2} + 5 x + 4\right) \cos{\left(x \right)}$}

\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}

HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (7 x^{2} + 5 x + 4)(\sin{\left(x \right)})(- 8 x^{2} - 8 x - 3) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%204%29%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%29%20" alt="LaTeX:  \displaystyle y = (7 x^{2} + 5 x + 4)(\sin{\left(x \right)})(- 8 x^{2} - 8 x - 3) " data-equation-content=" \displaystyle y = (7 x^{2} + 5 x + 4)(\sin{\left(x \right)})(- 8 x^{2} - 8 x - 3) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=7 x^{2} + 5 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B2%7D%20%2B%205%20x%20%2B%204%20" alt="LaTeX:  \displaystyle f=7 x^{2} + 5 x + 4 " data-equation-content=" \displaystyle f=7 x^{2} + 5 x + 4 " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=7 x^{2} + 5 x + 4 \implies f'=14 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B2%7D%20%2B%205%20x%20%2B%204%20%5Cimplies%20f%27%3D14%20x%20%2B%205%20" alt="LaTeX:  \displaystyle f=7 x^{2} + 5 x + 4 \implies f'=14 x + 5 " data-equation-content=" \displaystyle f=7 x^{2} + 5 x + 4 \implies f'=14 x + 5 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g = \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " data-equation-content=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " />  and  <img class="equation_image" title=" \displaystyle g=- 8 x^{2} - 8 x - 3 \implies g'=- 16 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%20%5Cimplies%20g%27%3D-%2016%20x%20-%208%20" alt="LaTeX:  \displaystyle g=- 8 x^{2} - 8 x - 3 \implies g'=- 16 x - 8 " data-equation-content=" \displaystyle g=- 8 x^{2} - 8 x - 3 \implies g'=- 16 x - 8 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 8 x^{2} - 8 x - 3)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 16 x - 8) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%2016%20x%20-%208%29%20" alt="LaTeX:  \displaystyle g'=(- 8 x^{2} - 8 x - 3)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 16 x - 8) " data-equation-content=" \displaystyle g'=(- 8 x^{2} - 8 x - 3)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 16 x - 8) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(7 x^{2} + 5 x + 4)(\left(- 16 x - 8\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \cos{\left(x \right)})+(\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)})(14 x + 5)=\left(- 16 x - 8\right) \left(7 x^{2} + 5 x + 4\right) \sin{\left(x \right)} + \left(14 x + 5\right) \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \left(7 x^{2} + 5 x + 4\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%204%29%28%5Cleft%28-%2016%20x%20-%208%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2814%20x%20%2B%205%29%3D%5Cleft%28-%2016%20x%20-%208%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%204%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2814%20x%20%2B%205%5Cright%29%20%5Cleft%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%208%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%204%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(7 x^{2} + 5 x + 4)(\left(- 16 x - 8\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \cos{\left(x \right)})+(\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)})(14 x + 5)=\left(- 16 x - 8\right) \left(7 x^{2} + 5 x + 4\right) \sin{\left(x \right)} + \left(14 x + 5\right) \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \left(7 x^{2} + 5 x + 4\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(7 x^{2} + 5 x + 4)(\left(- 16 x - 8\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \cos{\left(x \right)})+(\left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)})(14 x + 5)=\left(- 16 x - 8\right) \left(7 x^{2} + 5 x + 4\right) \sin{\left(x \right)} + \left(14 x + 5\right) \left(- 8 x^{2} - 8 x - 3\right) \sin{\left(x \right)} + \left(- 8 x^{2} - 8 x - 3\right) \left(7 x^{2} + 5 x + 4\right) \cos{\left(x \right)} " /> </p> </p>