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Identifying \(\displaystyle f=\cos{\left(x \right)}\) and \(\displaystyle g=\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right)\) and using the product rule with \(\displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)}\). This leaves g as \(\displaystyle g = \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x^{2} - 8 x - 4 \implies f'=18 x - 8\) and \(\displaystyle g=- 6 x^{2} + 9 x - 3 \implies g'=9 - 12 x\). Popping up a level gives \(\displaystyle g'=(- 6 x^{2} + 9 x - 3)(18 x - 8)+(9 x^{2} - 8 x - 4)(9 - 12 x)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(\cos{\left(x \right)})(\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right))+(\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right))(- \sin{\left(x \right)})=\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) \cos{\left(x \right)} + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right) \cos{\left(x \right)} - \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) \sin{\left(x \right)}\)

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\begin{question}Find the derivative of $y = (\cos{\left(x \right)})(9 x^{2} - 8 x - 4)(- 6 x^{2} + 9 x - 3)$.
    \soln{9cm}{Identifying $f=\cos{\left(x \right)}$ and $g=\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right)$ and using the product rule with $f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)}$. This leaves g as $g = \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x^{2} - 8 x - 4 \implies f'=18 x - 8$ and $g=- 6 x^{2} + 9 x - 3 \implies g'=9 - 12 x$. Popping up a level gives $g'=(- 6 x^{2} + 9 x - 3)(18 x - 8)+(9 x^{2} - 8 x - 4)(9 - 12 x)$Popping up again (Back to the original problem) gives $f'=(\cos{\left(x \right)})(\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right))+(\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right))(- \sin{\left(x \right)})=\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) \cos{\left(x \right)} + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right) \cos{\left(x \right)} - \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) \sin{\left(x \right)}$}

\end{question}

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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 = (\cos{\left(x \right)})(9 x^{2} - 8 x - 4)(- 6 x^{2} + 9 x - 3) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%289%20x%5E%7B2%7D%20-%208%20x%20-%204%29%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%29%20" alt="LaTeX:  \displaystyle y = (\cos{\left(x \right)})(9 x^{2} - 8 x - 4)(- 6 x^{2} + 9 x - 3) " data-equation-content=" \displaystyle y = (\cos{\left(x \right)})(9 x^{2} - 8 x - 4)(- 6 x^{2} + 9 x - 3) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f=\cos{\left(x \right)} " data-equation-content=" \displaystyle f=\cos{\left(x \right)} " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%20x%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) " data-equation-content=" \displaystyle g=\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)} " data-equation-content=" \displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)} " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%20x%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) " data-equation-content=" \displaystyle g = \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=9 x^{2} - 8 x - 4 \implies f'=18 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20-%208%20x%20-%204%20%5Cimplies%20f%27%3D18%20x%20-%208%20" alt="LaTeX:  \displaystyle f=9 x^{2} - 8 x - 4 \implies f'=18 x - 8 " data-equation-content=" \displaystyle f=9 x^{2} - 8 x - 4 \implies f'=18 x - 8 " />  and  <img class="equation_image" title=" \displaystyle g=- 6 x^{2} + 9 x - 3 \implies g'=9 - 12 x " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%20%5Cimplies%20g%27%3D9%20-%2012%20x%20" alt="LaTeX:  \displaystyle g=- 6 x^{2} + 9 x - 3 \implies g'=9 - 12 x " data-equation-content=" \displaystyle g=- 6 x^{2} + 9 x - 3 \implies g'=9 - 12 x " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 6 x^{2} + 9 x - 3)(18 x - 8)+(9 x^{2} - 8 x - 4)(9 - 12 x) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%29%2818%20x%20-%208%29%2B%289%20x%5E%7B2%7D%20-%208%20x%20-%204%29%289%20-%2012%20x%29%20" alt="LaTeX:  \displaystyle g'=(- 6 x^{2} + 9 x - 3)(18 x - 8)+(9 x^{2} - 8 x - 4)(9 - 12 x) " data-equation-content=" \displaystyle g'=(- 6 x^{2} + 9 x - 3)(18 x - 8)+(9 x^{2} - 8 x - 4)(9 - 12 x) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(\cos{\left(x \right)})(\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right))+(\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right))(- \sin{\left(x \right)})=\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) \cos{\left(x \right)} + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right) \cos{\left(x \right)} - \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Cleft%289%20-%2012%20x%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%20x%20-%204%5Cright%29%20%2B%20%5Cleft%2818%20x%20-%208%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%5Cright%29%29%2B%28%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%20x%20-%204%5Cright%29%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%3D%5Cleft%289%20-%2012%20x%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%20x%20-%204%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2818%20x%20-%208%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%209%20x%20-%203%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%20x%20-%204%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(\cos{\left(x \right)})(\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right))+(\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right))(- \sin{\left(x \right)})=\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) \cos{\left(x \right)} + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right) \cos{\left(x \right)} - \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle f'=(\cos{\left(x \right)})(\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right))+(\left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right))(- \sin{\left(x \right)})=\left(9 - 12 x\right) \left(9 x^{2} - 8 x - 4\right) \cos{\left(x \right)} + \left(18 x - 8\right) \left(- 6 x^{2} + 9 x - 3\right) \cos{\left(x \right)} - \left(- 6 x^{2} + 9 x - 3\right) \left(9 x^{2} - 8 x - 4\right) \sin{\left(x \right)} " /> </p> </p>