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Identifying \(\displaystyle f=7 x - 5\) and \(\displaystyle g=\left(2 x + 8\right) \left(9 x - 2\right)\) and using the product rule with \(\displaystyle f=7 x - 5 \implies f'=7\). This leaves g as \(\displaystyle g = \left(2 x + 8\right) \left(9 x - 2\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x - 2 \implies f'=9\) and \(\displaystyle g=2 x + 8 \implies g'=2\). Popping up a level gives \(\displaystyle g'=(2 x + 8)(9)+(9 x - 2)(2)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(7 x - 5)(36 x + 68)+(\left(2 x + 8\right) \left(9 x - 2\right))(7)=\left(2 x + 8\right) \left(63 x - 45\right) + \left(2 x + 8\right) \left(63 x - 14\right) + 2 \left(7 x - 5\right) \left(9 x - 2\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (7 x - 5)(9 x - 2)(2 x + 8)$.
    \soln{9cm}{Identifying $f=7 x - 5$ and $g=\left(2 x + 8\right) \left(9 x - 2\right)$ and using the product rule with $f=7 x - 5 \implies f'=7$. This leaves g as $g = \left(2 x + 8\right) \left(9 x - 2\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x - 2 \implies f'=9$ and $g=2 x + 8 \implies g'=2$. Popping up a level gives $g'=(2 x + 8)(9)+(9 x - 2)(2)$Popping up again (Back to the original problem) gives $f'=(7 x - 5)(36 x + 68)+(\left(2 x + 8\right) \left(9 x - 2\right))(7)=\left(2 x + 8\right) \left(63 x - 45\right) + \left(2 x + 8\right) \left(63 x - 14\right) + 2 \left(7 x - 5\right) \left(9 x - 2\right)$}

\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 x - 5)(9 x - 2)(2 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%287%20x%20-%205%29%289%20x%20-%202%29%282%20x%20%2B%208%29%20" alt="LaTeX:  \displaystyle y = (7 x - 5)(9 x - 2)(2 x + 8) " data-equation-content=" \displaystyle y = (7 x - 5)(9 x - 2)(2 x + 8) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=7 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%20-%205%20" alt="LaTeX:  \displaystyle f=7 x - 5 " data-equation-content=" \displaystyle f=7 x - 5 " />  and  <img class="equation_image" title=" \displaystyle g=\left(2 x + 8\right) \left(9 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%282%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(2 x + 8\right) \left(9 x - 2\right) " data-equation-content=" \displaystyle g=\left(2 x + 8\right) \left(9 x - 2\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=7 x - 5 \implies f'=7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%20-%205%20%5Cimplies%20f%27%3D7%20" alt="LaTeX:  \displaystyle f=7 x - 5 \implies f'=7 " data-equation-content=" \displaystyle f=7 x - 5 \implies f'=7 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(2 x + 8\right) \left(9 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%282%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(2 x + 8\right) \left(9 x - 2\right) " data-equation-content=" \displaystyle g = \left(2 x + 8\right) \left(9 x - 2\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=9 x - 2 \implies f'=9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%20-%202%20%5Cimplies%20f%27%3D9%20" alt="LaTeX:  \displaystyle f=9 x - 2 \implies f'=9 " data-equation-content=" \displaystyle f=9 x - 2 \implies f'=9 " />  and  <img class="equation_image" title=" \displaystyle g=2 x + 8 \implies g'=2 " src="/equation_images/%20%5Cdisplaystyle%20g%3D2%20x%20%2B%208%20%5Cimplies%20g%27%3D2%20" alt="LaTeX:  \displaystyle g=2 x + 8 \implies g'=2 " data-equation-content=" \displaystyle g=2 x + 8 \implies g'=2 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(2 x + 8)(9)+(9 x - 2)(2) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%282%20x%20%2B%208%29%289%29%2B%289%20x%20-%202%29%282%29%20" alt="LaTeX:  \displaystyle g'=(2 x + 8)(9)+(9 x - 2)(2) " data-equation-content=" \displaystyle g'=(2 x + 8)(9)+(9 x - 2)(2) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(7 x - 5)(36 x + 68)+(\left(2 x + 8\right) \left(9 x - 2\right))(7)=\left(2 x + 8\right) \left(63 x - 45\right) + \left(2 x + 8\right) \left(63 x - 14\right) + 2 \left(7 x - 5\right) \left(9 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%287%20x%20-%205%29%2836%20x%20%2B%2068%29%2B%28%5Cleft%282%20x%20%2B%208%5Cright%29%20%5Cleft%289%20x%20-%202%5Cright%29%29%287%29%3D%5Cleft%282%20x%20%2B%208%5Cright%29%20%5Cleft%2863%20x%20-%2045%5Cright%29%20%2B%20%5Cleft%282%20x%20%2B%208%5Cright%29%20%5Cleft%2863%20x%20-%2014%5Cright%29%20%2B%202%20%5Cleft%287%20x%20-%205%5Cright%29%20%5Cleft%289%20x%20-%202%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(7 x - 5)(36 x + 68)+(\left(2 x + 8\right) \left(9 x - 2\right))(7)=\left(2 x + 8\right) \left(63 x - 45\right) + \left(2 x + 8\right) \left(63 x - 14\right) + 2 \left(7 x - 5\right) \left(9 x - 2\right) " data-equation-content=" \displaystyle f'=(7 x - 5)(36 x + 68)+(\left(2 x + 8\right) \left(9 x - 2\right))(7)=\left(2 x + 8\right) \left(63 x - 45\right) + \left(2 x + 8\right) \left(63 x - 14\right) + 2 \left(7 x - 5\right) \left(9 x - 2\right) " /> </p> </p>