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Identifying \(\displaystyle f=9 - 7 x\) and \(\displaystyle g=\left(- 7 x - 9\right) \left(- 6 x - 4\right)\) and using the product rule with \(\displaystyle f=9 - 7 x \implies f'=-7\). This leaves g as \(\displaystyle g = \left(- 7 x - 9\right) \left(- 6 x - 4\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 6 x - 4 \implies f'=-6\) and \(\displaystyle g=- 7 x - 9 \implies g'=-7\). Popping up a level gives \(\displaystyle g'=(- 7 x - 9)(-6)+(- 6 x - 4)(-7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(9 - 7 x)(84 x + 82)+(\left(- 7 x - 9\right) \left(- 6 x - 4\right))(-7)=\left(9 - 7 x\right) \left(42 x + 28\right) + \left(9 - 7 x\right) \left(42 x + 54\right) - 7 \left(- 7 x - 9\right) \left(- 6 x - 4\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (9 - 7 x)(- 6 x - 4)(- 7 x - 9)$.
    \soln{9cm}{Identifying $f=9 - 7 x$ and $g=\left(- 7 x - 9\right) \left(- 6 x - 4\right)$ and using the product rule with $f=9 - 7 x \implies f'=-7$. This leaves g as $g = \left(- 7 x - 9\right) \left(- 6 x - 4\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 6 x - 4 \implies f'=-6$ and $g=- 7 x - 9 \implies g'=-7$. Popping up a level gives $g'=(- 7 x - 9)(-6)+(- 6 x - 4)(-7)$Popping up again (Back to the original problem) gives $f'=(9 - 7 x)(84 x + 82)+(\left(- 7 x - 9\right) \left(- 6 x - 4\right))(-7)=\left(9 - 7 x\right) \left(42 x + 28\right) + \left(9 - 7 x\right) \left(42 x + 54\right) - 7 \left(- 7 x - 9\right) \left(- 6 x - 4\right)$}

\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 = (9 - 7 x)(- 6 x - 4)(- 7 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%289%20-%207%20x%29%28-%206%20x%20-%204%29%28-%207%20x%20-%209%29%20" alt="LaTeX:  \displaystyle y = (9 - 7 x)(- 6 x - 4)(- 7 x - 9) " data-equation-content=" \displaystyle y = (9 - 7 x)(- 6 x - 4)(- 7 x - 9) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=9 - 7 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20-%207%20x%20" alt="LaTeX:  \displaystyle f=9 - 7 x " data-equation-content=" \displaystyle f=9 - 7 x " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 7 x - 9\right) \left(- 6 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%207%20x%20-%209%5Cright%29%20%5Cleft%28-%206%20x%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(- 7 x - 9\right) \left(- 6 x - 4\right) " data-equation-content=" \displaystyle g=\left(- 7 x - 9\right) \left(- 6 x - 4\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=9 - 7 x \implies f'=-7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20-%207%20x%20%5Cimplies%20f%27%3D-7%20" alt="LaTeX:  \displaystyle f=9 - 7 x \implies f'=-7 " data-equation-content=" \displaystyle f=9 - 7 x \implies f'=-7 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 7 x - 9\right) \left(- 6 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%207%20x%20-%209%5Cright%29%20%5Cleft%28-%206%20x%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(- 7 x - 9\right) \left(- 6 x - 4\right) " data-equation-content=" \displaystyle g = \left(- 7 x - 9\right) \left(- 6 x - 4\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=- 6 x - 4 \implies f'=-6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%20-%204%20%5Cimplies%20f%27%3D-6%20" alt="LaTeX:  \displaystyle f=- 6 x - 4 \implies f'=-6 " data-equation-content=" \displaystyle f=- 6 x - 4 \implies f'=-6 " />  and  <img class="equation_image" title=" \displaystyle g=- 7 x - 9 \implies g'=-7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%207%20x%20-%209%20%5Cimplies%20g%27%3D-7%20" alt="LaTeX:  \displaystyle g=- 7 x - 9 \implies g'=-7 " data-equation-content=" \displaystyle g=- 7 x - 9 \implies g'=-7 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 7 x - 9)(-6)+(- 6 x - 4)(-7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%207%20x%20-%209%29%28-6%29%2B%28-%206%20x%20-%204%29%28-7%29%20" alt="LaTeX:  \displaystyle g'=(- 7 x - 9)(-6)+(- 6 x - 4)(-7) " data-equation-content=" \displaystyle g'=(- 7 x - 9)(-6)+(- 6 x - 4)(-7) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(9 - 7 x)(84 x + 82)+(\left(- 7 x - 9\right) \left(- 6 x - 4\right))(-7)=\left(9 - 7 x\right) \left(42 x + 28\right) + \left(9 - 7 x\right) \left(42 x + 54\right) - 7 \left(- 7 x - 9\right) \left(- 6 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%289%20-%207%20x%29%2884%20x%20%2B%2082%29%2B%28%5Cleft%28-%207%20x%20-%209%5Cright%29%20%5Cleft%28-%206%20x%20-%204%5Cright%29%29%28-7%29%3D%5Cleft%289%20-%207%20x%5Cright%29%20%5Cleft%2842%20x%20%2B%2028%5Cright%29%20%2B%20%5Cleft%289%20-%207%20x%5Cright%29%20%5Cleft%2842%20x%20%2B%2054%5Cright%29%20-%207%20%5Cleft%28-%207%20x%20-%209%5Cright%29%20%5Cleft%28-%206%20x%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(9 - 7 x)(84 x + 82)+(\left(- 7 x - 9\right) \left(- 6 x - 4\right))(-7)=\left(9 - 7 x\right) \left(42 x + 28\right) + \left(9 - 7 x\right) \left(42 x + 54\right) - 7 \left(- 7 x - 9\right) \left(- 6 x - 4\right) " data-equation-content=" \displaystyle f'=(9 - 7 x)(84 x + 82)+(\left(- 7 x - 9\right) \left(- 6 x - 4\right))(-7)=\left(9 - 7 x\right) \left(42 x + 28\right) + \left(9 - 7 x\right) \left(42 x + 54\right) - 7 \left(- 7 x - 9\right) \left(- 6 x - 4\right) " /> </p> </p>