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Identifying \(\displaystyle f=2 x + 9\) and \(\displaystyle g=\left(9 - 9 x\right) \left(3 x + 9\right)\) and using the product rule with \(\displaystyle f=2 x + 9 \implies f'=2\). This leaves g as \(\displaystyle g = \left(9 - 9 x\right) \left(3 x + 9\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=3 x + 9 \implies f'=3\) and \(\displaystyle g=9 - 9 x \implies g'=-9\). Popping up a level gives \(\displaystyle g'=(9 - 9 x)(3)+(3 x + 9)(-9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(2 x + 9)(- 54 x - 54)+(\left(9 - 9 x\right) \left(3 x + 9\right))(2)=3 \left(9 - 9 x\right) \left(2 x + 9\right) + \left(18 - 18 x\right) \left(3 x + 9\right) + \left(- 18 x - 81\right) \left(3 x + 9\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (2 x + 9)(3 x + 9)(9 - 9 x)$.
    \soln{9cm}{Identifying $f=2 x + 9$ and $g=\left(9 - 9 x\right) \left(3 x + 9\right)$ and using the product rule with $f=2 x + 9 \implies f'=2$. This leaves g as $g = \left(9 - 9 x\right) \left(3 x + 9\right)$ which also requires the product rule. Pushing down in the new product rule $f=3 x + 9 \implies f'=3$ and $g=9 - 9 x \implies g'=-9$. Popping up a level gives $g'=(9 - 9 x)(3)+(3 x + 9)(-9)$Popping up again (Back to the original problem) gives $f'=(2 x + 9)(- 54 x - 54)+(\left(9 - 9 x\right) \left(3 x + 9\right))(2)=3 \left(9 - 9 x\right) \left(2 x + 9\right) + \left(18 - 18 x\right) \left(3 x + 9\right) + \left(- 18 x - 81\right) \left(3 x + 9\right)$}

\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 = (2 x + 9)(3 x + 9)(9 - 9 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%282%20x%20%2B%209%29%283%20x%20%2B%209%29%289%20-%209%20x%29%20" alt="LaTeX:  \displaystyle y = (2 x + 9)(3 x + 9)(9 - 9 x) " data-equation-content=" \displaystyle y = (2 x + 9)(3 x + 9)(9 - 9 x) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=2 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20x%20%2B%209%20" alt="LaTeX:  \displaystyle f=2 x + 9 " data-equation-content=" \displaystyle f=2 x + 9 " />  and  <img class="equation_image" title=" \displaystyle g=\left(9 - 9 x\right) \left(3 x + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%289%20-%209%20x%5Cright%29%20%5Cleft%283%20x%20%2B%209%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(9 - 9 x\right) \left(3 x + 9\right) " data-equation-content=" \displaystyle g=\left(9 - 9 x\right) \left(3 x + 9\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=2 x + 9 \implies f'=2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20x%20%2B%209%20%5Cimplies%20f%27%3D2%20" alt="LaTeX:  \displaystyle f=2 x + 9 \implies f'=2 " data-equation-content=" \displaystyle f=2 x + 9 \implies f'=2 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(9 - 9 x\right) \left(3 x + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%289%20-%209%20x%5Cright%29%20%5Cleft%283%20x%20%2B%209%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(9 - 9 x\right) \left(3 x + 9\right) " data-equation-content=" \displaystyle g = \left(9 - 9 x\right) \left(3 x + 9\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=3 x + 9 \implies f'=3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20x%20%2B%209%20%5Cimplies%20f%27%3D3%20" alt="LaTeX:  \displaystyle f=3 x + 9 \implies f'=3 " data-equation-content=" \displaystyle f=3 x + 9 \implies f'=3 " />  and  <img class="equation_image" title=" \displaystyle g=9 - 9 x \implies g'=-9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D9%20-%209%20x%20%5Cimplies%20g%27%3D-9%20" alt="LaTeX:  \displaystyle g=9 - 9 x \implies g'=-9 " data-equation-content=" \displaystyle g=9 - 9 x \implies g'=-9 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(9 - 9 x)(3)+(3 x + 9)(-9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%289%20-%209%20x%29%283%29%2B%283%20x%20%2B%209%29%28-9%29%20" alt="LaTeX:  \displaystyle g'=(9 - 9 x)(3)+(3 x + 9)(-9) " data-equation-content=" \displaystyle g'=(9 - 9 x)(3)+(3 x + 9)(-9) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(2 x + 9)(- 54 x - 54)+(\left(9 - 9 x\right) \left(3 x + 9\right))(2)=3 \left(9 - 9 x\right) \left(2 x + 9\right) + \left(18 - 18 x\right) \left(3 x + 9\right) + \left(- 18 x - 81\right) \left(3 x + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%282%20x%20%2B%209%29%28-%2054%20x%20-%2054%29%2B%28%5Cleft%289%20-%209%20x%5Cright%29%20%5Cleft%283%20x%20%2B%209%5Cright%29%29%282%29%3D3%20%5Cleft%289%20-%209%20x%5Cright%29%20%5Cleft%282%20x%20%2B%209%5Cright%29%20%2B%20%5Cleft%2818%20-%2018%20x%5Cright%29%20%5Cleft%283%20x%20%2B%209%5Cright%29%20%2B%20%5Cleft%28-%2018%20x%20-%2081%5Cright%29%20%5Cleft%283%20x%20%2B%209%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(2 x + 9)(- 54 x - 54)+(\left(9 - 9 x\right) \left(3 x + 9\right))(2)=3 \left(9 - 9 x\right) \left(2 x + 9\right) + \left(18 - 18 x\right) \left(3 x + 9\right) + \left(- 18 x - 81\right) \left(3 x + 9\right) " data-equation-content=" \displaystyle f'=(2 x + 9)(- 54 x - 54)+(\left(9 - 9 x\right) \left(3 x + 9\right))(2)=3 \left(9 - 9 x\right) \left(2 x + 9\right) + \left(18 - 18 x\right) \left(3 x + 9\right) + \left(- 18 x - 81\right) \left(3 x + 9\right) " /> </p> </p>