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Identifying \(\displaystyle f=9 x + 6\) and \(\displaystyle g=\left(4 - 9 x\right) \sin{\left(x \right)}\) and using the product rule with \(\displaystyle f=9 x + 6 \implies f'=9\). This leaves g as \(\displaystyle g = \left(4 - 9 x\right) \sin{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=4 - 9 x \implies f'=-9\) and \(\displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)}\). Popping up a level gives \(\displaystyle g'=(\sin{\left(x \right)})(-9)+(4 - 9 x)(\cos{\left(x \right)})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(9 x + 6)(\left(4 - 9 x\right) \cos{\left(x \right)} - 9 \sin{\left(x \right)})+(\left(4 - 9 x\right) \sin{\left(x \right)})(9)=\left(4 - 9 x\right) \left(9 x + 6\right) \cos{\left(x \right)} + \left(36 - 81 x\right) \sin{\left(x \right)} + \left(- 81 x - 54\right) \sin{\left(x \right)}\)

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\begin{question}Find the derivative of $y = (9 x + 6)(4 - 9 x)(\sin{\left(x \right)})$.
    \soln{9cm}{Identifying $f=9 x + 6$ and $g=\left(4 - 9 x\right) \sin{\left(x \right)}$ and using the product rule with $f=9 x + 6 \implies f'=9$. This leaves g as $g = \left(4 - 9 x\right) \sin{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=4 - 9 x \implies f'=-9$ and $g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)}$. Popping up a level gives $g'=(\sin{\left(x \right)})(-9)+(4 - 9 x)(\cos{\left(x \right)})$Popping up again (Back to the original problem) gives $f'=(9 x + 6)(\left(4 - 9 x\right) \cos{\left(x \right)} - 9 \sin{\left(x \right)})+(\left(4 - 9 x\right) \sin{\left(x \right)})(9)=\left(4 - 9 x\right) \left(9 x + 6\right) \cos{\left(x \right)} + \left(36 - 81 x\right) \sin{\left(x \right)} + \left(- 81 x - 54\right) \sin{\left(x \right)}$}

\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.}

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HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (9 x + 6)(4 - 9 x)(\sin{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%289%20x%20%2B%206%29%284%20-%209%20x%29%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle y = (9 x + 6)(4 - 9 x)(\sin{\left(x \right)}) " data-equation-content=" \displaystyle y = (9 x + 6)(4 - 9 x)(\sin{\left(x \right)}) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=9 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%20%2B%206%20" alt="LaTeX:  \displaystyle f=9 x + 6 " data-equation-content=" \displaystyle f=9 x + 6 " />  and  <img class="equation_image" title=" \displaystyle g=\left(4 - 9 x\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%284%20-%209%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(4 - 9 x\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\left(4 - 9 x\right) \sin{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=9 x + 6 \implies f'=9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%20%2B%206%20%5Cimplies%20f%27%3D9%20" alt="LaTeX:  \displaystyle f=9 x + 6 \implies f'=9 " data-equation-content=" \displaystyle f=9 x + 6 \implies f'=9 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(4 - 9 x\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%284%20-%209%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(4 - 9 x\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g = \left(4 - 9 x\right) \sin{\left(x \right)} " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=4 - 9 x \implies f'=-9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20-%209%20x%20%5Cimplies%20f%27%3D-9%20" alt="LaTeX:  \displaystyle f=4 - 9 x \implies f'=-9 " data-equation-content=" \displaystyle f=4 - 9 x \implies f'=-9 " />  and  <img class="equation_image" title=" \displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)} " data-equation-content=" \displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)} " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(\sin{\left(x \right)})(-9)+(4 - 9 x)(\cos{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-9%29%2B%284%20-%209%20x%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle g'=(\sin{\left(x \right)})(-9)+(4 - 9 x)(\cos{\left(x \right)}) " data-equation-content=" \displaystyle g'=(\sin{\left(x \right)})(-9)+(4 - 9 x)(\cos{\left(x \right)}) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(9 x + 6)(\left(4 - 9 x\right) \cos{\left(x \right)} - 9 \sin{\left(x \right)})+(\left(4 - 9 x\right) \sin{\left(x \right)})(9)=\left(4 - 9 x\right) \left(9 x + 6\right) \cos{\left(x \right)} + \left(36 - 81 x\right) \sin{\left(x \right)} + \left(- 81 x - 54\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%289%20x%20%2B%206%29%28%5Cleft%284%20-%209%20x%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%209%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%284%20-%209%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%289%29%3D%5Cleft%284%20-%209%20x%5Cright%29%20%5Cleft%289%20x%20%2B%206%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2836%20-%2081%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%2081%20x%20-%2054%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(9 x + 6)(\left(4 - 9 x\right) \cos{\left(x \right)} - 9 \sin{\left(x \right)})+(\left(4 - 9 x\right) \sin{\left(x \right)})(9)=\left(4 - 9 x\right) \left(9 x + 6\right) \cos{\left(x \right)} + \left(36 - 81 x\right) \sin{\left(x \right)} + \left(- 81 x - 54\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle f'=(9 x + 6)(\left(4 - 9 x\right) \cos{\left(x \right)} - 9 \sin{\left(x \right)})+(\left(4 - 9 x\right) \sin{\left(x \right)})(9)=\left(4 - 9 x\right) \left(9 x + 6\right) \cos{\left(x \right)} + \left(36 - 81 x\right) \sin{\left(x \right)} + \left(- 81 x - 54\right) \sin{\left(x \right)} " /> </p> </p>