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Identifying \(\displaystyle f=\sin{\left(x \right)}\) and \(\displaystyle g=\left(4 - 7 x\right) \left(- 4 x - 5\right)\) and using the product rule with \(\displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}\). This leaves g as \(\displaystyle g = \left(4 - 7 x\right) \left(- 4 x - 5\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 4 x - 5 \implies f'=-4\) and \(\displaystyle g=4 - 7 x \implies g'=-7\). Popping up a level gives \(\displaystyle g'=(4 - 7 x)(-4)+(- 4 x - 5)(-7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(\sin{\left(x \right)})(56 x + 19)+(\left(4 - 7 x\right) \left(- 4 x - 5\right))(\cos{\left(x \right)})=\left(4 - 7 x\right) \left(- 4 x - 5\right) \cos{\left(x \right)} + \left(28 x - 16\right) \sin{\left(x \right)} + \left(28 x + 35\right) \sin{\left(x \right)}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (\sin{\left(x \right)})(- 4 x - 5)(4 - 7 x)$.
    \soln{9cm}{Identifying $f=\sin{\left(x \right)}$ and $g=\left(4 - 7 x\right) \left(- 4 x - 5\right)$ and using the product rule with $f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}$. This leaves g as $g = \left(4 - 7 x\right) \left(- 4 x - 5\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 4 x - 5 \implies f'=-4$ and $g=4 - 7 x \implies g'=-7$. Popping up a level gives $g'=(4 - 7 x)(-4)+(- 4 x - 5)(-7)$Popping up again (Back to the original problem) gives $f'=(\sin{\left(x \right)})(56 x + 19)+(\left(4 - 7 x\right) \left(- 4 x - 5\right))(\cos{\left(x \right)})=\left(4 - 7 x\right) \left(- 4 x - 5\right) \cos{\left(x \right)} + \left(28 x - 16\right) \sin{\left(x \right)} + \left(28 x + 35\right) \sin{\left(x \right)}$}

\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 = (\sin{\left(x \right)})(- 4 x - 5)(4 - 7 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%204%20x%20-%205%29%284%20-%207%20x%29%20" alt="LaTeX:  \displaystyle y = (\sin{\left(x \right)})(- 4 x - 5)(4 - 7 x) " data-equation-content=" \displaystyle y = (\sin{\left(x \right)})(- 4 x - 5)(4 - 7 x) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=\sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f=\sin{\left(x \right)} " data-equation-content=" \displaystyle f=\sin{\left(x \right)} " />  and  <img class="equation_image" title=" \displaystyle g=\left(4 - 7 x\right) \left(- 4 x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%284%20-%207%20x%5Cright%29%20%5Cleft%28-%204%20x%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(4 - 7 x\right) \left(- 4 x - 5\right) " data-equation-content=" \displaystyle g=\left(4 - 7 x\right) \left(- 4 x - 5\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " data-equation-content=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(4 - 7 x\right) \left(- 4 x - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%284%20-%207%20x%5Cright%29%20%5Cleft%28-%204%20x%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(4 - 7 x\right) \left(- 4 x - 5\right) " data-equation-content=" \displaystyle g = \left(4 - 7 x\right) \left(- 4 x - 5\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=- 4 x - 5 \implies f'=-4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%20-%205%20%5Cimplies%20f%27%3D-4%20" alt="LaTeX:  \displaystyle f=- 4 x - 5 \implies f'=-4 " data-equation-content=" \displaystyle f=- 4 x - 5 \implies f'=-4 " />  and  <img class="equation_image" title=" \displaystyle g=4 - 7 x \implies g'=-7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D4%20-%207%20x%20%5Cimplies%20g%27%3D-7%20" alt="LaTeX:  \displaystyle g=4 - 7 x \implies g'=-7 " data-equation-content=" \displaystyle g=4 - 7 x \implies g'=-7 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(4 - 7 x)(-4)+(- 4 x - 5)(-7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%284%20-%207%20x%29%28-4%29%2B%28-%204%20x%20-%205%29%28-7%29%20" alt="LaTeX:  \displaystyle g'=(4 - 7 x)(-4)+(- 4 x - 5)(-7) " data-equation-content=" \displaystyle g'=(4 - 7 x)(-4)+(- 4 x - 5)(-7) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(\sin{\left(x \right)})(56 x + 19)+(\left(4 - 7 x\right) \left(- 4 x - 5\right))(\cos{\left(x \right)})=\left(4 - 7 x\right) \left(- 4 x - 5\right) \cos{\left(x \right)} + \left(28 x - 16\right) \sin{\left(x \right)} + \left(28 x + 35\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2856%20x%20%2B%2019%29%2B%28%5Cleft%284%20-%207%20x%5Cright%29%20%5Cleft%28-%204%20x%20-%205%5Cright%29%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%3D%5Cleft%284%20-%207%20x%5Cright%29%20%5Cleft%28-%204%20x%20-%205%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2828%20x%20-%2016%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2828%20x%20%2B%2035%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(\sin{\left(x \right)})(56 x + 19)+(\left(4 - 7 x\right) \left(- 4 x - 5\right))(\cos{\left(x \right)})=\left(4 - 7 x\right) \left(- 4 x - 5\right) \cos{\left(x \right)} + \left(28 x - 16\right) \sin{\left(x \right)} + \left(28 x + 35\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle f'=(\sin{\left(x \right)})(56 x + 19)+(\left(4 - 7 x\right) \left(- 4 x - 5\right))(\cos{\left(x \right)})=\left(4 - 7 x\right) \left(- 4 x - 5\right) \cos{\left(x \right)} + \left(28 x - 16\right) \sin{\left(x \right)} + \left(28 x + 35\right) \sin{\left(x \right)} " /> </p> </p>