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Identifying \(\displaystyle f=\sin{\left(x \right)}\) and \(\displaystyle g=\left(1 - 5 x\right) \left(7 x + 1\right)\) and using the product rule with \(\displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}\). This leaves g as \(\displaystyle g = \left(1 - 5 x\right) \left(7 x + 1\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=7 x + 1 \implies f'=7\) and \(\displaystyle g=1 - 5 x \implies g'=-5\). Popping up a level gives \(\displaystyle g'=(1 - 5 x)(7)+(7 x + 1)(-5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(\sin{\left(x \right)})(2 - 70 x)+(\left(1 - 5 x\right) \left(7 x + 1\right))(\cos{\left(x \right)})=\left(1 - 5 x\right) \left(7 x + 1\right) \cos{\left(x \right)} + \left(7 - 35 x\right) \sin{\left(x \right)} + \left(- 35 x - 5\right) \sin{\left(x \right)}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (\sin{\left(x \right)})(7 x + 1)(1 - 5 x)$.
    \soln{9cm}{Identifying $f=\sin{\left(x \right)}$ and $g=\left(1 - 5 x\right) \left(7 x + 1\right)$ and using the product rule with $f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}$. This leaves g as $g = \left(1 - 5 x\right) \left(7 x + 1\right)$ which also requires the product rule. Pushing down in the new product rule $f=7 x + 1 \implies f'=7$ and $g=1 - 5 x \implies g'=-5$. Popping up a level gives $g'=(1 - 5 x)(7)+(7 x + 1)(-5)$Popping up again (Back to the original problem) gives $f'=(\sin{\left(x \right)})(2 - 70 x)+(\left(1 - 5 x\right) \left(7 x + 1\right))(\cos{\left(x \right)})=\left(1 - 5 x\right) \left(7 x + 1\right) \cos{\left(x \right)} + \left(7 - 35 x\right) \sin{\left(x \right)} + \left(- 35 x - 5\right) \sin{\left(x \right)}$}

\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)})(7 x + 1)(1 - 5 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%287%20x%20%2B%201%29%281%20-%205%20x%29%20" alt="LaTeX:  \displaystyle y = (\sin{\left(x \right)})(7 x + 1)(1 - 5 x) " data-equation-content=" \displaystyle y = (\sin{\left(x \right)})(7 x + 1)(1 - 5 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(1 - 5 x\right) \left(7 x + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%281%20-%205%20x%5Cright%29%20%5Cleft%287%20x%20%2B%201%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(1 - 5 x\right) \left(7 x + 1\right) " data-equation-content=" \displaystyle g=\left(1 - 5 x\right) \left(7 x + 1\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(1 - 5 x\right) \left(7 x + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%281%20-%205%20x%5Cright%29%20%5Cleft%287%20x%20%2B%201%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(1 - 5 x\right) \left(7 x + 1\right) " data-equation-content=" \displaystyle g = \left(1 - 5 x\right) \left(7 x + 1\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=7 x + 1 \implies f'=7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%20%2B%201%20%5Cimplies%20f%27%3D7%20" alt="LaTeX:  \displaystyle f=7 x + 1 \implies f'=7 " data-equation-content=" \displaystyle f=7 x + 1 \implies f'=7 " />  and  <img class="equation_image" title=" \displaystyle g=1 - 5 x \implies g'=-5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D1%20-%205%20x%20%5Cimplies%20g%27%3D-5%20" alt="LaTeX:  \displaystyle g=1 - 5 x \implies g'=-5 " data-equation-content=" \displaystyle g=1 - 5 x \implies g'=-5 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(1 - 5 x)(7)+(7 x + 1)(-5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%281%20-%205%20x%29%287%29%2B%287%20x%20%2B%201%29%28-5%29%20" alt="LaTeX:  \displaystyle g'=(1 - 5 x)(7)+(7 x + 1)(-5) " data-equation-content=" \displaystyle g'=(1 - 5 x)(7)+(7 x + 1)(-5) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(\sin{\left(x \right)})(2 - 70 x)+(\left(1 - 5 x\right) \left(7 x + 1\right))(\cos{\left(x \right)})=\left(1 - 5 x\right) \left(7 x + 1\right) \cos{\left(x \right)} + \left(7 - 35 x\right) \sin{\left(x \right)} + \left(- 35 x - 5\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%282%20-%2070%20x%29%2B%28%5Cleft%281%20-%205%20x%5Cright%29%20%5Cleft%287%20x%20%2B%201%5Cright%29%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%3D%5Cleft%281%20-%205%20x%5Cright%29%20%5Cleft%287%20x%20%2B%201%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%287%20-%2035%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%2035%20x%20-%205%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(\sin{\left(x \right)})(2 - 70 x)+(\left(1 - 5 x\right) \left(7 x + 1\right))(\cos{\left(x \right)})=\left(1 - 5 x\right) \left(7 x + 1\right) \cos{\left(x \right)} + \left(7 - 35 x\right) \sin{\left(x \right)} + \left(- 35 x - 5\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle f'=(\sin{\left(x \right)})(2 - 70 x)+(\left(1 - 5 x\right) \left(7 x + 1\right))(\cos{\left(x \right)})=\left(1 - 5 x\right) \left(7 x + 1\right) \cos{\left(x \right)} + \left(7 - 35 x\right) \sin{\left(x \right)} + \left(- 35 x - 5\right) \sin{\left(x \right)} " /> </p> </p>