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Identifying \(\displaystyle f=\log{\left(x \right)}\) and \(\displaystyle g=\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)\) and using the product rule with \(\displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x}\). This leaves g as \(\displaystyle g = \left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 9 x^{2} + 6 x - 7 \implies f'=6 - 18 x\) and \(\displaystyle g=- 6 x^{2} + 2 x + 4 \implies g'=2 - 12 x\). Popping up a level gives \(\displaystyle g'=(- 6 x^{2} + 2 x + 4)(6 - 18 x)+(- 9 x^{2} + 6 x - 7)(2 - 12 x)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(\log{\left(x \right)})(\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right))+(\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right))(\frac{1}{x})=\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) \log{\left(x \right)} + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)}{x}\)

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\begin{question}Find the derivative of $y = (\log{\left(x \right)})(- 9 x^{2} + 6 x - 7)(- 6 x^{2} + 2 x + 4)$.
    \soln{9cm}{Identifying $f=\log{\left(x \right)}$ and $g=\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)$ and using the product rule with $f=\log{\left(x \right)} \implies f'=\frac{1}{x}$. This leaves g as $g = \left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 9 x^{2} + 6 x - 7 \implies f'=6 - 18 x$ and $g=- 6 x^{2} + 2 x + 4 \implies g'=2 - 12 x$. Popping up a level gives $g'=(- 6 x^{2} + 2 x + 4)(6 - 18 x)+(- 9 x^{2} + 6 x - 7)(2 - 12 x)$Popping up again (Back to the original problem) gives $f'=(\log{\left(x \right)})(\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right))+(\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right))(\frac{1}{x})=\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) \log{\left(x \right)} + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)}{x}$}

\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 = (\log{\left(x \right)})(- 9 x^{2} + 6 x - 7)(- 6 x^{2} + 2 x + 4) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%29%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%29%20" alt="LaTeX:  \displaystyle y = (\log{\left(x \right)})(- 9 x^{2} + 6 x - 7)(- 6 x^{2} + 2 x + 4) " data-equation-content=" \displaystyle y = (\log{\left(x \right)})(- 9 x^{2} + 6 x - 7)(- 6 x^{2} + 2 x + 4) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=\log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f=\log{\left(x \right)} " data-equation-content=" \displaystyle f=\log{\left(x \right)} " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right) " data-equation-content=" \displaystyle g=\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " data-equation-content=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right) " data-equation-content=" \displaystyle g = \left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=- 9 x^{2} + 6 x - 7 \implies f'=6 - 18 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%20%5Cimplies%20f%27%3D6%20-%2018%20x%20" alt="LaTeX:  \displaystyle f=- 9 x^{2} + 6 x - 7 \implies f'=6 - 18 x " data-equation-content=" \displaystyle f=- 9 x^{2} + 6 x - 7 \implies f'=6 - 18 x " />  and  <img class="equation_image" title=" \displaystyle g=- 6 x^{2} + 2 x + 4 \implies g'=2 - 12 x " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%20%5Cimplies%20g%27%3D2%20-%2012%20x%20" alt="LaTeX:  \displaystyle g=- 6 x^{2} + 2 x + 4 \implies g'=2 - 12 x " data-equation-content=" \displaystyle g=- 6 x^{2} + 2 x + 4 \implies g'=2 - 12 x " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 6 x^{2} + 2 x + 4)(6 - 18 x)+(- 9 x^{2} + 6 x - 7)(2 - 12 x) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%29%286%20-%2018%20x%29%2B%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%29%282%20-%2012%20x%29%20" alt="LaTeX:  \displaystyle g'=(- 6 x^{2} + 2 x + 4)(6 - 18 x)+(- 9 x^{2} + 6 x - 7)(2 - 12 x) " data-equation-content=" \displaystyle g'=(- 6 x^{2} + 2 x + 4)(6 - 18 x)+(- 9 x^{2} + 6 x - 7)(2 - 12 x) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(\log{\left(x \right)})(\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right))+(\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right))(\frac{1}{x})=\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) \log{\left(x \right)} + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Cleft%282%20-%2012%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%5Cright%29%20%2B%20%5Cleft%286%20-%2018%20x%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%29%2B%28%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%29%28%5Cfrac%7B1%7D%7Bx%7D%29%3D%5Cleft%282%20-%2012%20x%5Cright%29%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%286%20-%2018%20x%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%206%20x%20-%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%202%20x%20%2B%204%5Cright%29%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(\log{\left(x \right)})(\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right))+(\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right))(\frac{1}{x})=\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) \log{\left(x \right)} + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)}{x} " data-equation-content=" \displaystyle f'=(\log{\left(x \right)})(\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right))+(\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right))(\frac{1}{x})=\left(2 - 12 x\right) \left(- 9 x^{2} + 6 x - 7\right) \log{\left(x \right)} + \left(6 - 18 x\right) \left(- 6 x^{2} + 2 x + 4\right) \log{\left(x \right)} + \frac{\left(- 9 x^{2} + 6 x - 7\right) \left(- 6 x^{2} + 2 x + 4\right)}{x} " /> </p> </p>