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Identifying \(\displaystyle f=- 9 x - 7\) and \(\displaystyle g=\left(- 5 x - 9\right) \cos{\left(x \right)}\) and using the product rule with \(\displaystyle f=- 9 x - 7 \implies f'=-9\). This leaves g as \(\displaystyle g = \left(- 5 x - 9\right) \cos{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 5 x - 9 \implies f'=-5\) and \(\displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)}\). Popping up a level gives \(\displaystyle g'=(\cos{\left(x \right)})(-5)+(- 5 x - 9)(- \sin{\left(x \right)})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 9 x - 7)(- \left(- 5 x - 9\right) \sin{\left(x \right)} - 5 \cos{\left(x \right)})+(\left(- 5 x - 9\right) \cos{\left(x \right)})(-9)=- \left(- 9 x - 7\right) \left(- 5 x - 9\right) \sin{\left(x \right)} + \left(45 x + 35\right) \cos{\left(x \right)} + \left(45 x + 81\right) \cos{\left(x \right)}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (- 9 x - 7)(- 5 x - 9)(\cos{\left(x \right)})$.
    \soln{9cm}{Identifying $f=- 9 x - 7$ and $g=\left(- 5 x - 9\right) \cos{\left(x \right)}$ and using the product rule with $f=- 9 x - 7 \implies f'=-9$. This leaves g as $g = \left(- 5 x - 9\right) \cos{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=- 5 x - 9 \implies f'=-5$ and $g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)}$. Popping up a level gives $g'=(\cos{\left(x \right)})(-5)+(- 5 x - 9)(- \sin{\left(x \right)})$Popping up again (Back to the original problem) gives $f'=(- 9 x - 7)(- \left(- 5 x - 9\right) \sin{\left(x \right)} - 5 \cos{\left(x \right)})+(\left(- 5 x - 9\right) \cos{\left(x \right)})(-9)=- \left(- 9 x - 7\right) \left(- 5 x - 9\right) \sin{\left(x \right)} + \left(45 x + 35\right) \cos{\left(x \right)} + \left(45 x + 81\right) \cos{\left(x \right)}$}

\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 - 7)(- 5 x - 9)(\cos{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%209%20x%20-%207%29%28-%205%20x%20-%209%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle y = (- 9 x - 7)(- 5 x - 9)(\cos{\left(x \right)}) " data-equation-content=" \displaystyle y = (- 9 x - 7)(- 5 x - 9)(\cos{\left(x \right)}) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 9 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%209%20x%20-%207%20" alt="LaTeX:  \displaystyle f=- 9 x - 7 " data-equation-content=" \displaystyle f=- 9 x - 7 " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 5 x - 9\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%205%20x%20-%209%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(- 5 x - 9\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g=\left(- 5 x - 9\right) \cos{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 9 x - 7 \implies f'=-9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%209%20x%20-%207%20%5Cimplies%20f%27%3D-9%20" alt="LaTeX:  \displaystyle f=- 9 x - 7 \implies f'=-9 " data-equation-content=" \displaystyle f=- 9 x - 7 \implies f'=-9 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 5 x - 9\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%205%20x%20-%209%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(- 5 x - 9\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g = \left(- 5 x - 9\right) \cos{\left(x \right)} " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=- 5 x - 9 \implies f'=-5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%20-%209%20%5Cimplies%20f%27%3D-5%20" alt="LaTeX:  \displaystyle f=- 5 x - 9 \implies f'=-5 " data-equation-content=" \displaystyle f=- 5 x - 9 \implies f'=-5 " />  and  <img class="equation_image" title=" \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(\cos{\left(x \right)})(-5)+(- 5 x - 9)(- \sin{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28-5%29%2B%28-%205%20x%20-%209%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle g'=(\cos{\left(x \right)})(-5)+(- 5 x - 9)(- \sin{\left(x \right)}) " data-equation-content=" \displaystyle g'=(\cos{\left(x \right)})(-5)+(- 5 x - 9)(- \sin{\left(x \right)}) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 9 x - 7)(- \left(- 5 x - 9\right) \sin{\left(x \right)} - 5 \cos{\left(x \right)})+(\left(- 5 x - 9\right) \cos{\left(x \right)})(-9)=- \left(- 9 x - 7\right) \left(- 5 x - 9\right) \sin{\left(x \right)} + \left(45 x + 35\right) \cos{\left(x \right)} + \left(45 x + 81\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%209%20x%20-%207%29%28-%20%5Cleft%28-%205%20x%20-%209%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%205%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%28-%205%20x%20-%209%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28-9%29%3D-%20%5Cleft%28-%209%20x%20-%207%5Cright%29%20%5Cleft%28-%205%20x%20-%209%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2845%20x%20%2B%2035%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2845%20x%20%2B%2081%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(- 9 x - 7)(- \left(- 5 x - 9\right) \sin{\left(x \right)} - 5 \cos{\left(x \right)})+(\left(- 5 x - 9\right) \cos{\left(x \right)})(-9)=- \left(- 9 x - 7\right) \left(- 5 x - 9\right) \sin{\left(x \right)} + \left(45 x + 35\right) \cos{\left(x \right)} + \left(45 x + 81\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(- 9 x - 7)(- \left(- 5 x - 9\right) \sin{\left(x \right)} - 5 \cos{\left(x \right)})+(\left(- 5 x - 9\right) \cos{\left(x \right)})(-9)=- \left(- 9 x - 7\right) \left(- 5 x - 9\right) \sin{\left(x \right)} + \left(45 x + 35\right) \cos{\left(x \right)} + \left(45 x + 81\right) \cos{\left(x \right)} " /> </p> </p>