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Identifying \(\displaystyle f=- 3 x - 6\) and \(\displaystyle g=\left(3 x - 5\right) \left(4 x + 7\right)\) and using the product rule with \(\displaystyle f=- 3 x - 6 \implies f'=-3\). This leaves g as \(\displaystyle g = \left(3 x - 5\right) \left(4 x + 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=4 x + 7 \implies f'=4\) and \(\displaystyle g=3 x - 5 \implies g'=3\). Popping up a level gives \(\displaystyle g'=(3 x - 5)(4)+(4 x + 7)(3)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 3 x - 6)(24 x + 1)+(\left(3 x - 5\right) \left(4 x + 7\right))(-3)=\left(15 - 9 x\right) \left(4 x + 7\right) + \left(- 9 x - 18\right) \left(4 x + 7\right) + 4 \left(- 3 x - 6\right) \left(3 x - 5\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (- 3 x - 6)(4 x + 7)(3 x - 5)$.
    \soln{9cm}{Identifying $f=- 3 x - 6$ and $g=\left(3 x - 5\right) \left(4 x + 7\right)$ and using the product rule with $f=- 3 x - 6 \implies f'=-3$. This leaves g as $g = \left(3 x - 5\right) \left(4 x + 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=4 x + 7 \implies f'=4$ and $g=3 x - 5 \implies g'=3$. Popping up a level gives $g'=(3 x - 5)(4)+(4 x + 7)(3)$Popping up again (Back to the original problem) gives $f'=(- 3 x - 6)(24 x + 1)+(\left(3 x - 5\right) \left(4 x + 7\right))(-3)=\left(15 - 9 x\right) \left(4 x + 7\right) + \left(- 9 x - 18\right) \left(4 x + 7\right) + 4 \left(- 3 x - 6\right) \left(3 x - 5\right)$}

\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 = (- 3 x - 6)(4 x + 7)(3 x - 5) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%203%20x%20-%206%29%284%20x%20%2B%207%29%283%20x%20-%205%29%20" alt="LaTeX:  \displaystyle y = (- 3 x - 6)(4 x + 7)(3 x - 5) " data-equation-content=" \displaystyle y = (- 3 x - 6)(4 x + 7)(3 x - 5) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 3 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%20-%206%20" alt="LaTeX:  \displaystyle f=- 3 x - 6 " data-equation-content=" \displaystyle f=- 3 x - 6 " />  and  <img class="equation_image" title=" \displaystyle g=\left(3 x - 5\right) \left(4 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%283%20x%20-%205%5Cright%29%20%5Cleft%284%20x%20%2B%207%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(3 x - 5\right) \left(4 x + 7\right) " data-equation-content=" \displaystyle g=\left(3 x - 5\right) \left(4 x + 7\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 3 x - 6 \implies f'=-3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%20-%206%20%5Cimplies%20f%27%3D-3%20" alt="LaTeX:  \displaystyle f=- 3 x - 6 \implies f'=-3 " data-equation-content=" \displaystyle f=- 3 x - 6 \implies f'=-3 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(3 x - 5\right) \left(4 x + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%283%20x%20-%205%5Cright%29%20%5Cleft%284%20x%20%2B%207%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(3 x - 5\right) \left(4 x + 7\right) " data-equation-content=" \displaystyle g = \left(3 x - 5\right) \left(4 x + 7\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=4 x + 7 \implies f'=4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20x%20%2B%207%20%5Cimplies%20f%27%3D4%20" alt="LaTeX:  \displaystyle f=4 x + 7 \implies f'=4 " data-equation-content=" \displaystyle f=4 x + 7 \implies f'=4 " />  and  <img class="equation_image" title=" \displaystyle g=3 x - 5 \implies g'=3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D3%20x%20-%205%20%5Cimplies%20g%27%3D3%20" alt="LaTeX:  \displaystyle g=3 x - 5 \implies g'=3 " data-equation-content=" \displaystyle g=3 x - 5 \implies g'=3 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(3 x - 5)(4)+(4 x + 7)(3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%283%20x%20-%205%29%284%29%2B%284%20x%20%2B%207%29%283%29%20" alt="LaTeX:  \displaystyle g'=(3 x - 5)(4)+(4 x + 7)(3) " data-equation-content=" \displaystyle g'=(3 x - 5)(4)+(4 x + 7)(3) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 3 x - 6)(24 x + 1)+(\left(3 x - 5\right) \left(4 x + 7\right))(-3)=\left(15 - 9 x\right) \left(4 x + 7\right) + \left(- 9 x - 18\right) \left(4 x + 7\right) + 4 \left(- 3 x - 6\right) \left(3 x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%203%20x%20-%206%29%2824%20x%20%2B%201%29%2B%28%5Cleft%283%20x%20-%205%5Cright%29%20%5Cleft%284%20x%20%2B%207%5Cright%29%29%28-3%29%3D%5Cleft%2815%20-%209%20x%5Cright%29%20%5Cleft%284%20x%20%2B%207%5Cright%29%20%2B%20%5Cleft%28-%209%20x%20-%2018%5Cright%29%20%5Cleft%284%20x%20%2B%207%5Cright%29%20%2B%204%20%5Cleft%28-%203%20x%20-%206%5Cright%29%20%5Cleft%283%20x%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(- 3 x - 6)(24 x + 1)+(\left(3 x - 5\right) \left(4 x + 7\right))(-3)=\left(15 - 9 x\right) \left(4 x + 7\right) + \left(- 9 x - 18\right) \left(4 x + 7\right) + 4 \left(- 3 x - 6\right) \left(3 x - 5\right) " data-equation-content=" \displaystyle f'=(- 3 x - 6)(24 x + 1)+(\left(3 x - 5\right) \left(4 x + 7\right))(-3)=\left(15 - 9 x\right) \left(4 x + 7\right) + \left(- 9 x - 18\right) \left(4 x + 7\right) + 4 \left(- 3 x - 6\right) \left(3 x - 5\right) " /> </p> </p>