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Identifying \(\displaystyle f=2 x + 3\) and \(\displaystyle g=\left(5 - 6 x\right) \left(7 x - 7\right)\) and using the product rule with \(\displaystyle f=2 x + 3 \implies f'=2\). This leaves g as \(\displaystyle g = \left(5 - 6 x\right) \left(7 x - 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=7 x - 7 \implies f'=7\) and \(\displaystyle g=5 - 6 x \implies g'=-6\). Popping up a level gives \(\displaystyle g'=(5 - 6 x)(7)+(7 x - 7)(-6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(2 x + 3)(77 - 84 x)+(\left(5 - 6 x\right) \left(7 x - 7\right))(2)=\left(5 - 6 x\right) \left(14 x - 14\right) + \left(5 - 6 x\right) \left(14 x + 21\right) - 6 \left(2 x + 3\right) \left(7 x - 7\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (2 x + 3)(7 x - 7)(5 - 6 x)$.
    \soln{9cm}{Identifying $f=2 x + 3$ and $g=\left(5 - 6 x\right) \left(7 x - 7\right)$ and using the product rule with $f=2 x + 3 \implies f'=2$. This leaves g as $g = \left(5 - 6 x\right) \left(7 x - 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=7 x - 7 \implies f'=7$ and $g=5 - 6 x \implies g'=-6$. Popping up a level gives $g'=(5 - 6 x)(7)+(7 x - 7)(-6)$Popping up again (Back to the original problem) gives $f'=(2 x + 3)(77 - 84 x)+(\left(5 - 6 x\right) \left(7 x - 7\right))(2)=\left(5 - 6 x\right) \left(14 x - 14\right) + \left(5 - 6 x\right) \left(14 x + 21\right) - 6 \left(2 x + 3\right) \left(7 x - 7\right)$}

\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 + 3)(7 x - 7)(5 - 6 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%282%20x%20%2B%203%29%287%20x%20-%207%29%285%20-%206%20x%29%20" alt="LaTeX:  \displaystyle y = (2 x + 3)(7 x - 7)(5 - 6 x) " data-equation-content=" \displaystyle y = (2 x + 3)(7 x - 7)(5 - 6 x) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=2 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20x%20%2B%203%20" alt="LaTeX:  \displaystyle f=2 x + 3 " data-equation-content=" \displaystyle f=2 x + 3 " />  and  <img class="equation_image" title=" \displaystyle g=\left(5 - 6 x\right) \left(7 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%285%20-%206%20x%5Cright%29%20%5Cleft%287%20x%20-%207%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(5 - 6 x\right) \left(7 x - 7\right) " data-equation-content=" \displaystyle g=\left(5 - 6 x\right) \left(7 x - 7\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=2 x + 3 \implies f'=2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20x%20%2B%203%20%5Cimplies%20f%27%3D2%20" alt="LaTeX:  \displaystyle f=2 x + 3 \implies f'=2 " data-equation-content=" \displaystyle f=2 x + 3 \implies f'=2 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(5 - 6 x\right) \left(7 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%285%20-%206%20x%5Cright%29%20%5Cleft%287%20x%20-%207%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(5 - 6 x\right) \left(7 x - 7\right) " data-equation-content=" \displaystyle g = \left(5 - 6 x\right) \left(7 x - 7\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=7 x - 7 \implies f'=7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%20-%207%20%5Cimplies%20f%27%3D7%20" alt="LaTeX:  \displaystyle f=7 x - 7 \implies f'=7 " data-equation-content=" \displaystyle f=7 x - 7 \implies f'=7 " />  and  <img class="equation_image" title=" \displaystyle g=5 - 6 x \implies g'=-6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D5%20-%206%20x%20%5Cimplies%20g%27%3D-6%20" alt="LaTeX:  \displaystyle g=5 - 6 x \implies g'=-6 " data-equation-content=" \displaystyle g=5 - 6 x \implies g'=-6 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(5 - 6 x)(7)+(7 x - 7)(-6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%285%20-%206%20x%29%287%29%2B%287%20x%20-%207%29%28-6%29%20" alt="LaTeX:  \displaystyle g'=(5 - 6 x)(7)+(7 x - 7)(-6) " data-equation-content=" \displaystyle g'=(5 - 6 x)(7)+(7 x - 7)(-6) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(2 x + 3)(77 - 84 x)+(\left(5 - 6 x\right) \left(7 x - 7\right))(2)=\left(5 - 6 x\right) \left(14 x - 14\right) + \left(5 - 6 x\right) \left(14 x + 21\right) - 6 \left(2 x + 3\right) \left(7 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%282%20x%20%2B%203%29%2877%20-%2084%20x%29%2B%28%5Cleft%285%20-%206%20x%5Cright%29%20%5Cleft%287%20x%20-%207%5Cright%29%29%282%29%3D%5Cleft%285%20-%206%20x%5Cright%29%20%5Cleft%2814%20x%20-%2014%5Cright%29%20%2B%20%5Cleft%285%20-%206%20x%5Cright%29%20%5Cleft%2814%20x%20%2B%2021%5Cright%29%20-%206%20%5Cleft%282%20x%20%2B%203%5Cright%29%20%5Cleft%287%20x%20-%207%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(2 x + 3)(77 - 84 x)+(\left(5 - 6 x\right) \left(7 x - 7\right))(2)=\left(5 - 6 x\right) \left(14 x - 14\right) + \left(5 - 6 x\right) \left(14 x + 21\right) - 6 \left(2 x + 3\right) \left(7 x - 7\right) " data-equation-content=" \displaystyle f'=(2 x + 3)(77 - 84 x)+(\left(5 - 6 x\right) \left(7 x - 7\right))(2)=\left(5 - 6 x\right) \left(14 x - 14\right) + \left(5 - 6 x\right) \left(14 x + 21\right) - 6 \left(2 x + 3\right) \left(7 x - 7\right) " /> </p> </p>