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Identifying \(\displaystyle f=- 5 x - 6\) and \(\displaystyle g=\left(8 - 7 x\right) \cos{\left(x \right)}\) and using the product rule with \(\displaystyle f=- 5 x - 6 \implies f'=-5\). This leaves g as \(\displaystyle g = \left(8 - 7 x\right) \cos{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)}\) and \(\displaystyle g=8 - 7 x \implies g'=-7\). Popping up a level gives \(\displaystyle g'=(8 - 7 x)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(-7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 5 x - 6)(- \left(8 - 7 x\right) \sin{\left(x \right)} - 7 \cos{\left(x \right)})+(\left(8 - 7 x\right) \cos{\left(x \right)})(-5)=- \left(8 - 7 x\right) \left(- 5 x - 6\right) \sin{\left(x \right)} + \left(35 x - 40\right) \cos{\left(x \right)} + \left(35 x + 42\right) \cos{\left(x \right)}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (- 5 x - 6)(\cos{\left(x \right)})(8 - 7 x)$.
    \soln{9cm}{Identifying $f=- 5 x - 6$ and $g=\left(8 - 7 x\right) \cos{\left(x \right)}$ and using the product rule with $f=- 5 x - 6 \implies f'=-5$. This leaves g as $g = \left(8 - 7 x\right) \cos{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\cos{\left(x \right)} \implies f'=- \sin{\left(x \right)}$ and $g=8 - 7 x \implies g'=-7$. Popping up a level gives $g'=(8 - 7 x)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(-7)$Popping up again (Back to the original problem) gives $f'=(- 5 x - 6)(- \left(8 - 7 x\right) \sin{\left(x \right)} - 7 \cos{\left(x \right)})+(\left(8 - 7 x\right) \cos{\left(x \right)})(-5)=- \left(8 - 7 x\right) \left(- 5 x - 6\right) \sin{\left(x \right)} + \left(35 x - 40\right) \cos{\left(x \right)} + \left(35 x + 42\right) \cos{\left(x \right)}$}

\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 = (- 5 x - 6)(\cos{\left(x \right)})(8 - 7 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%205%20x%20-%206%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%288%20-%207%20x%29%20" alt="LaTeX:  \displaystyle y = (- 5 x - 6)(\cos{\left(x \right)})(8 - 7 x) " data-equation-content=" \displaystyle y = (- 5 x - 6)(\cos{\left(x \right)})(8 - 7 x) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 5 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%20-%206%20" alt="LaTeX:  \displaystyle f=- 5 x - 6 " data-equation-content=" \displaystyle f=- 5 x - 6 " />  and  <img class="equation_image" title=" \displaystyle g=\left(8 - 7 x\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%288%20-%207%20x%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(8 - 7 x\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g=\left(8 - 7 x\right) \cos{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 5 x - 6 \implies f'=-5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%20-%206%20%5Cimplies%20f%27%3D-5%20" alt="LaTeX:  \displaystyle f=- 5 x - 6 \implies f'=-5 " data-equation-content=" \displaystyle f=- 5 x - 6 \implies f'=-5 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(8 - 7 x\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%288%20-%207%20x%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(8 - 7 x\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g = \left(8 - 7 x\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=\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)} " />  and  <img class="equation_image" title=" \displaystyle g=8 - 7 x \implies g'=-7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D8%20-%207%20x%20%5Cimplies%20g%27%3D-7%20" alt="LaTeX:  \displaystyle g=8 - 7 x \implies g'=-7 " data-equation-content=" \displaystyle g=8 - 7 x \implies g'=-7 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(8 - 7 x)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(-7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%288%20-%207%20x%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28-7%29%20" alt="LaTeX:  \displaystyle g'=(8 - 7 x)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(-7) " data-equation-content=" \displaystyle g'=(8 - 7 x)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(-7) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 5 x - 6)(- \left(8 - 7 x\right) \sin{\left(x \right)} - 7 \cos{\left(x \right)})+(\left(8 - 7 x\right) \cos{\left(x \right)})(-5)=- \left(8 - 7 x\right) \left(- 5 x - 6\right) \sin{\left(x \right)} + \left(35 x - 40\right) \cos{\left(x \right)} + \left(35 x + 42\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%205%20x%20-%206%29%28-%20%5Cleft%288%20-%207%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%207%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%288%20-%207%20x%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28-5%29%3D-%20%5Cleft%288%20-%207%20x%5Cright%29%20%5Cleft%28-%205%20x%20-%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2835%20x%20-%2040%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2835%20x%20%2B%2042%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(- 5 x - 6)(- \left(8 - 7 x\right) \sin{\left(x \right)} - 7 \cos{\left(x \right)})+(\left(8 - 7 x\right) \cos{\left(x \right)})(-5)=- \left(8 - 7 x\right) \left(- 5 x - 6\right) \sin{\left(x \right)} + \left(35 x - 40\right) \cos{\left(x \right)} + \left(35 x + 42\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(- 5 x - 6)(- \left(8 - 7 x\right) \sin{\left(x \right)} - 7 \cos{\left(x \right)})+(\left(8 - 7 x\right) \cos{\left(x \right)})(-5)=- \left(8 - 7 x\right) \left(- 5 x - 6\right) \sin{\left(x \right)} + \left(35 x - 40\right) \cos{\left(x \right)} + \left(35 x + 42\right) \cos{\left(x \right)} " /> </p> </p>