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Identifying \(\displaystyle f=3 - 3 x\) and \(\displaystyle g=\left(4 - 9 x\right) \left(5 x + 5\right)\) and using the product rule with \(\displaystyle f=3 - 3 x \implies f'=-3\). This leaves g as \(\displaystyle g = \left(4 - 9 x\right) \left(5 x + 5\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=5 x + 5 \implies f'=5\) and \(\displaystyle g=4 - 9 x \implies g'=-9\). Popping up a level gives \(\displaystyle g'=(4 - 9 x)(5)+(5 x + 5)(-9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(3 - 3 x)(- 90 x - 25)+(\left(4 - 9 x\right) \left(5 x + 5\right))(-3)=5 \left(3 - 3 x\right) \left(4 - 9 x\right) + \left(5 x + 5\right) \left(27 x - 27\right) + \left(5 x + 5\right) \left(27 x - 12\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (3 - 3 x)(5 x + 5)(4 - 9 x)$.
    \soln{9cm}{Identifying $f=3 - 3 x$ and $g=\left(4 - 9 x\right) \left(5 x + 5\right)$ and using the product rule with $f=3 - 3 x \implies f'=-3$. This leaves g as $g = \left(4 - 9 x\right) \left(5 x + 5\right)$ which also requires the product rule. Pushing down in the new product rule $f=5 x + 5 \implies f'=5$ and $g=4 - 9 x \implies g'=-9$. Popping up a level gives $g'=(4 - 9 x)(5)+(5 x + 5)(-9)$Popping up again (Back to the original problem) gives $f'=(3 - 3 x)(- 90 x - 25)+(\left(4 - 9 x\right) \left(5 x + 5\right))(-3)=5 \left(3 - 3 x\right) \left(4 - 9 x\right) + \left(5 x + 5\right) \left(27 x - 27\right) + \left(5 x + 5\right) \left(27 x - 12\right)$}

\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 - 3 x)(5 x + 5)(4 - 9 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%283%20-%203%20x%29%285%20x%20%2B%205%29%284%20-%209%20x%29%20" alt="LaTeX:  \displaystyle y = (3 - 3 x)(5 x + 5)(4 - 9 x) " data-equation-content=" \displaystyle y = (3 - 3 x)(5 x + 5)(4 - 9 x) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=3 - 3 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20-%203%20x%20" alt="LaTeX:  \displaystyle f=3 - 3 x " data-equation-content=" \displaystyle f=3 - 3 x " />  and  <img class="equation_image" title=" \displaystyle g=\left(4 - 9 x\right) \left(5 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%284%20-%209%20x%5Cright%29%20%5Cleft%285%20x%20%2B%205%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(4 - 9 x\right) \left(5 x + 5\right) " data-equation-content=" \displaystyle g=\left(4 - 9 x\right) \left(5 x + 5\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=3 - 3 x \implies f'=-3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20-%203%20x%20%5Cimplies%20f%27%3D-3%20" alt="LaTeX:  \displaystyle f=3 - 3 x \implies f'=-3 " data-equation-content=" \displaystyle f=3 - 3 x \implies f'=-3 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(4 - 9 x\right) \left(5 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%284%20-%209%20x%5Cright%29%20%5Cleft%285%20x%20%2B%205%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(4 - 9 x\right) \left(5 x + 5\right) " data-equation-content=" \displaystyle g = \left(4 - 9 x\right) \left(5 x + 5\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=5 x + 5 \implies f'=5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%20%2B%205%20%5Cimplies%20f%27%3D5%20" alt="LaTeX:  \displaystyle f=5 x + 5 \implies f'=5 " data-equation-content=" \displaystyle f=5 x + 5 \implies f'=5 " />  and  <img class="equation_image" title=" \displaystyle g=4 - 9 x \implies g'=-9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D4%20-%209%20x%20%5Cimplies%20g%27%3D-9%20" alt="LaTeX:  \displaystyle g=4 - 9 x \implies g'=-9 " data-equation-content=" \displaystyle g=4 - 9 x \implies g'=-9 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(4 - 9 x)(5)+(5 x + 5)(-9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%284%20-%209%20x%29%285%29%2B%285%20x%20%2B%205%29%28-9%29%20" alt="LaTeX:  \displaystyle g'=(4 - 9 x)(5)+(5 x + 5)(-9) " data-equation-content=" \displaystyle g'=(4 - 9 x)(5)+(5 x + 5)(-9) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(3 - 3 x)(- 90 x - 25)+(\left(4 - 9 x\right) \left(5 x + 5\right))(-3)=5 \left(3 - 3 x\right) \left(4 - 9 x\right) + \left(5 x + 5\right) \left(27 x - 27\right) + \left(5 x + 5\right) \left(27 x - 12\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%283%20-%203%20x%29%28-%2090%20x%20-%2025%29%2B%28%5Cleft%284%20-%209%20x%5Cright%29%20%5Cleft%285%20x%20%2B%205%5Cright%29%29%28-3%29%3D5%20%5Cleft%283%20-%203%20x%5Cright%29%20%5Cleft%284%20-%209%20x%5Cright%29%20%2B%20%5Cleft%285%20x%20%2B%205%5Cright%29%20%5Cleft%2827%20x%20-%2027%5Cright%29%20%2B%20%5Cleft%285%20x%20%2B%205%5Cright%29%20%5Cleft%2827%20x%20-%2012%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(3 - 3 x)(- 90 x - 25)+(\left(4 - 9 x\right) \left(5 x + 5\right))(-3)=5 \left(3 - 3 x\right) \left(4 - 9 x\right) + \left(5 x + 5\right) \left(27 x - 27\right) + \left(5 x + 5\right) \left(27 x - 12\right) " data-equation-content=" \displaystyle f'=(3 - 3 x)(- 90 x - 25)+(\left(4 - 9 x\right) \left(5 x + 5\right))(-3)=5 \left(3 - 3 x\right) \left(4 - 9 x\right) + \left(5 x + 5\right) \left(27 x - 27\right) + \left(5 x + 5\right) \left(27 x - 12\right) " /> </p> </p>