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

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

\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 = (- 3 x^{2} + 4 x + 3)(- 9 x^{2} - 9 x - 7)(\log{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%203%20x%5E%7B2%7D%20%2B%204%20x%20%2B%203%29%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle y = (- 3 x^{2} + 4 x + 3)(- 9 x^{2} - 9 x - 7)(\log{\left(x \right)}) " data-equation-content=" \displaystyle y = (- 3 x^{2} + 4 x + 3)(- 9 x^{2} - 9 x - 7)(\log{\left(x \right)}) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 3 x^{2} + 4 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%5E%7B2%7D%20%2B%204%20x%20%2B%203%20" alt="LaTeX:  \displaystyle f=- 3 x^{2} + 4 x + 3 " data-equation-content=" \displaystyle f=- 3 x^{2} + 4 x + 3 " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g=\left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 3 x^{2} + 4 x + 3 \implies f'=4 - 6 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%5E%7B2%7D%20%2B%204%20x%20%2B%203%20%5Cimplies%20f%27%3D4%20-%206%20x%20" alt="LaTeX:  \displaystyle f=- 3 x^{2} + 4 x + 3 \implies f'=4 - 6 x " data-equation-content=" \displaystyle f=- 3 x^{2} + 4 x + 3 \implies f'=4 - 6 x " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g = \left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=- 9 x^{2} - 9 x - 7 \implies f'=- 18 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%20%5Cimplies%20f%27%3D-%2018%20x%20-%209%20" alt="LaTeX:  \displaystyle f=- 9 x^{2} - 9 x - 7 \implies f'=- 18 x - 9 " data-equation-content=" \displaystyle f=- 9 x^{2} - 9 x - 7 \implies f'=- 18 x - 9 " />  and  <img class="equation_image" title=" \displaystyle g=\log{\left(x \right)} \implies g'=\frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle g=\log{\left(x \right)} \implies g'=\frac{1}{x} " data-equation-content=" \displaystyle g=\log{\left(x \right)} \implies g'=\frac{1}{x} " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(\log{\left(x \right)})(- 18 x - 9)+(- 9 x^{2} - 9 x - 7)(\frac{1}{x}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%2018%20x%20-%209%29%2B%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%29%28%5Cfrac%7B1%7D%7Bx%7D%29%20" alt="LaTeX:  \displaystyle g'=(\log{\left(x \right)})(- 18 x - 9)+(- 9 x^{2} - 9 x - 7)(\frac{1}{x}) " data-equation-content=" \displaystyle g'=(\log{\left(x \right)})(- 18 x - 9)+(- 9 x^{2} - 9 x - 7)(\frac{1}{x}) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 3 x^{2} + 4 x + 3)(\left(- 18 x - 9\right) \log{\left(x \right)} + \frac{- 9 x^{2} - 9 x - 7}{x})+(\left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)})(4 - 6 x)=\left(4 - 6 x\right) \left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} + \left(- 18 x - 9\right) \left(- 3 x^{2} + 4 x + 3\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} - 9 x - 7\right) \left(- 3 x^{2} + 4 x + 3\right)}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%203%20x%5E%7B2%7D%20%2B%204%20x%20%2B%203%29%28%5Cleft%28-%2018%20x%20-%209%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%7D%7Bx%7D%29%2B%28%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%284%20-%206%20x%29%3D%5Cleft%284%20-%206%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%2018%20x%20-%209%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20%2B%204%20x%20%2B%203%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Cleft%28-%209%20x%5E%7B2%7D%20-%209%20x%20-%207%5Cright%29%20%5Cleft%28-%203%20x%5E%7B2%7D%20%2B%204%20x%20%2B%203%5Cright%29%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(- 3 x^{2} + 4 x + 3)(\left(- 18 x - 9\right) \log{\left(x \right)} + \frac{- 9 x^{2} - 9 x - 7}{x})+(\left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)})(4 - 6 x)=\left(4 - 6 x\right) \left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} + \left(- 18 x - 9\right) \left(- 3 x^{2} + 4 x + 3\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} - 9 x - 7\right) \left(- 3 x^{2} + 4 x + 3\right)}{x} " data-equation-content=" \displaystyle f'=(- 3 x^{2} + 4 x + 3)(\left(- 18 x - 9\right) \log{\left(x \right)} + \frac{- 9 x^{2} - 9 x - 7}{x})+(\left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)})(4 - 6 x)=\left(4 - 6 x\right) \left(- 9 x^{2} - 9 x - 7\right) \log{\left(x \right)} + \left(- 18 x - 9\right) \left(- 3 x^{2} + 4 x + 3\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} - 9 x - 7\right) \left(- 3 x^{2} + 4 x + 3\right)}{x} " /> </p> </p>