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

Download \(\LaTeX\)

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

\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 = (7 - 7 x)(3 x - 2)(- 3 x - 1) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%287%20-%207%20x%29%283%20x%20-%202%29%28-%203%20x%20-%201%29%20" alt="LaTeX:  \displaystyle y = (7 - 7 x)(3 x - 2)(- 3 x - 1) " data-equation-content=" \displaystyle y = (7 - 7 x)(3 x - 2)(- 3 x - 1) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=7 - 7 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20-%207%20x%20" alt="LaTeX:  \displaystyle f=7 - 7 x " data-equation-content=" \displaystyle f=7 - 7 x " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 3 x - 1\right) \left(3 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%203%20x%20-%201%5Cright%29%20%5Cleft%283%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(- 3 x - 1\right) \left(3 x - 2\right) " data-equation-content=" \displaystyle g=\left(- 3 x - 1\right) \left(3 x - 2\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=7 - 7 x \implies f'=-7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20-%207%20x%20%5Cimplies%20f%27%3D-7%20" alt="LaTeX:  \displaystyle f=7 - 7 x \implies f'=-7 " data-equation-content=" \displaystyle f=7 - 7 x \implies f'=-7 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 3 x - 1\right) \left(3 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%203%20x%20-%201%5Cright%29%20%5Cleft%283%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(- 3 x - 1\right) \left(3 x - 2\right) " data-equation-content=" \displaystyle g = \left(- 3 x - 1\right) \left(3 x - 2\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=3 x - 2 \implies f'=3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20x%20-%202%20%5Cimplies%20f%27%3D3%20" alt="LaTeX:  \displaystyle f=3 x - 2 \implies f'=3 " data-equation-content=" \displaystyle f=3 x - 2 \implies f'=3 " />  and  <img class="equation_image" title=" \displaystyle g=- 3 x - 1 \implies g'=-3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%203%20x%20-%201%20%5Cimplies%20g%27%3D-3%20" alt="LaTeX:  \displaystyle g=- 3 x - 1 \implies g'=-3 " data-equation-content=" \displaystyle g=- 3 x - 1 \implies g'=-3 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 3 x - 1)(3)+(3 x - 2)(-3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%203%20x%20-%201%29%283%29%2B%283%20x%20-%202%29%28-3%29%20" alt="LaTeX:  \displaystyle g'=(- 3 x - 1)(3)+(3 x - 2)(-3) " data-equation-content=" \displaystyle g'=(- 3 x - 1)(3)+(3 x - 2)(-3) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(7 - 7 x)(3 - 18 x)+(\left(- 3 x - 1\right) \left(3 x - 2\right))(-7)=\left(6 - 9 x\right) \left(7 - 7 x\right) + \left(7 - 7 x\right) \left(- 9 x - 3\right) - 7 \left(- 3 x - 1\right) \left(3 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%287%20-%207%20x%29%283%20-%2018%20x%29%2B%28%5Cleft%28-%203%20x%20-%201%5Cright%29%20%5Cleft%283%20x%20-%202%5Cright%29%29%28-7%29%3D%5Cleft%286%20-%209%20x%5Cright%29%20%5Cleft%287%20-%207%20x%5Cright%29%20%2B%20%5Cleft%287%20-%207%20x%5Cright%29%20%5Cleft%28-%209%20x%20-%203%5Cright%29%20-%207%20%5Cleft%28-%203%20x%20-%201%5Cright%29%20%5Cleft%283%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(7 - 7 x)(3 - 18 x)+(\left(- 3 x - 1\right) \left(3 x - 2\right))(-7)=\left(6 - 9 x\right) \left(7 - 7 x\right) + \left(7 - 7 x\right) \left(- 9 x - 3\right) - 7 \left(- 3 x - 1\right) \left(3 x - 2\right) " data-equation-content=" \displaystyle f'=(7 - 7 x)(3 - 18 x)+(\left(- 3 x - 1\right) \left(3 x - 2\right))(-7)=\left(6 - 9 x\right) \left(7 - 7 x\right) + \left(7 - 7 x\right) \left(- 9 x - 3\right) - 7 \left(- 3 x - 1\right) \left(3 x - 2\right) " /> </p> </p>