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Identifying \(\displaystyle f=5 x - 8\) and \(\displaystyle g=\left(2 x + 4\right) \left(3 x - 6\right)\) and using the product rule with \(\displaystyle f=5 x - 8 \implies f'=5\). This leaves g as \(\displaystyle g = \left(2 x + 4\right) \left(3 x - 6\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=2 x + 4 \implies f'=2\) and \(\displaystyle g=3 x - 6 \implies g'=3\). Popping up a level gives \(\displaystyle g'=(3 x - 6)(2)+(2 x + 4)(3)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(5 x - 8)(12 x)+(\left(2 x + 4\right) \left(3 x - 6\right))(5)=\left(2 x + 4\right) \left(15 x - 30\right) + \left(2 x + 4\right) \left(15 x - 24\right) + 2 \left(3 x - 6\right) \left(5 x - 8\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (5 x - 8)(2 x + 4)(3 x - 6)$.
    \soln{9cm}{Identifying $f=5 x - 8$ and $g=\left(2 x + 4\right) \left(3 x - 6\right)$ and using the product rule with $f=5 x - 8 \implies f'=5$. This leaves g as $g = \left(2 x + 4\right) \left(3 x - 6\right)$ which also requires the product rule. Pushing down in the new product rule $f=2 x + 4 \implies f'=2$ and $g=3 x - 6 \implies g'=3$. Popping up a level gives $g'=(3 x - 6)(2)+(2 x + 4)(3)$Popping up again (Back to the original problem) gives $f'=(5 x - 8)(12 x)+(\left(2 x + 4\right) \left(3 x - 6\right))(5)=\left(2 x + 4\right) \left(15 x - 30\right) + \left(2 x + 4\right) \left(15 x - 24\right) + 2 \left(3 x - 6\right) \left(5 x - 8\right)$}

\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 = (5 x - 8)(2 x + 4)(3 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%285%20x%20-%208%29%282%20x%20%2B%204%29%283%20x%20-%206%29%20" alt="LaTeX:  \displaystyle y = (5 x - 8)(2 x + 4)(3 x - 6) " data-equation-content=" \displaystyle y = (5 x - 8)(2 x + 4)(3 x - 6) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=5 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%20-%208%20" alt="LaTeX:  \displaystyle f=5 x - 8 " data-equation-content=" \displaystyle f=5 x - 8 " />  and  <img class="equation_image" title=" \displaystyle g=\left(2 x + 4\right) \left(3 x - 6\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%282%20x%20%2B%204%5Cright%29%20%5Cleft%283%20x%20-%206%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(2 x + 4\right) \left(3 x - 6\right) " data-equation-content=" \displaystyle g=\left(2 x + 4\right) \left(3 x - 6\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=5 x - 8 \implies f'=5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%20-%208%20%5Cimplies%20f%27%3D5%20" alt="LaTeX:  \displaystyle f=5 x - 8 \implies f'=5 " data-equation-content=" \displaystyle f=5 x - 8 \implies f'=5 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(2 x + 4\right) \left(3 x - 6\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%282%20x%20%2B%204%5Cright%29%20%5Cleft%283%20x%20-%206%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(2 x + 4\right) \left(3 x - 6\right) " data-equation-content=" \displaystyle g = \left(2 x + 4\right) \left(3 x - 6\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=2 x + 4 \implies f'=2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20x%20%2B%204%20%5Cimplies%20f%27%3D2%20" alt="LaTeX:  \displaystyle f=2 x + 4 \implies f'=2 " data-equation-content=" \displaystyle f=2 x + 4 \implies f'=2 " />  and  <img class="equation_image" title=" \displaystyle g=3 x - 6 \implies g'=3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D3%20x%20-%206%20%5Cimplies%20g%27%3D3%20" alt="LaTeX:  \displaystyle g=3 x - 6 \implies g'=3 " data-equation-content=" \displaystyle g=3 x - 6 \implies g'=3 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(3 x - 6)(2)+(2 x + 4)(3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%283%20x%20-%206%29%282%29%2B%282%20x%20%2B%204%29%283%29%20" alt="LaTeX:  \displaystyle g'=(3 x - 6)(2)+(2 x + 4)(3) " data-equation-content=" \displaystyle g'=(3 x - 6)(2)+(2 x + 4)(3) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(5 x - 8)(12 x)+(\left(2 x + 4\right) \left(3 x - 6\right))(5)=\left(2 x + 4\right) \left(15 x - 30\right) + \left(2 x + 4\right) \left(15 x - 24\right) + 2 \left(3 x - 6\right) \left(5 x - 8\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%285%20x%20-%208%29%2812%20x%29%2B%28%5Cleft%282%20x%20%2B%204%5Cright%29%20%5Cleft%283%20x%20-%206%5Cright%29%29%285%29%3D%5Cleft%282%20x%20%2B%204%5Cright%29%20%5Cleft%2815%20x%20-%2030%5Cright%29%20%2B%20%5Cleft%282%20x%20%2B%204%5Cright%29%20%5Cleft%2815%20x%20-%2024%5Cright%29%20%2B%202%20%5Cleft%283%20x%20-%206%5Cright%29%20%5Cleft%285%20x%20-%208%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(5 x - 8)(12 x)+(\left(2 x + 4\right) \left(3 x - 6\right))(5)=\left(2 x + 4\right) \left(15 x - 30\right) + \left(2 x + 4\right) \left(15 x - 24\right) + 2 \left(3 x - 6\right) \left(5 x - 8\right) " data-equation-content=" \displaystyle f'=(5 x - 8)(12 x)+(\left(2 x + 4\right) \left(3 x - 6\right))(5)=\left(2 x + 4\right) \left(15 x - 30\right) + \left(2 x + 4\right) \left(15 x - 24\right) + 2 \left(3 x - 6\right) \left(5 x - 8\right) " /> </p> </p>