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Identifying \(\displaystyle f=5 x + 6\) and \(\displaystyle g=\left(5 x + 4\right) \cos{\left(x \right)}\) and using the product rule with \(\displaystyle f=5 x + 6 \implies f'=5\). This leaves g as \(\displaystyle g = \left(5 x + 4\right) \cos{\left(x \right)}\) 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 \(\displaystyle g=5 x + 4 \implies g'=5\). Popping up a level gives \(\displaystyle g'=(5 x + 4)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(5 x + 6)(- \left(5 x + 4\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)})+(\left(5 x + 4\right) \cos{\left(x \right)})(5)=- \left(5 x + 4\right) \left(5 x + 6\right) \sin{\left(x \right)} + \left(25 x + 20\right) \cos{\left(x \right)} + \left(25 x + 30\right) \cos{\left(x \right)}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (5 x + 6)(\cos{\left(x \right)})(5 x + 4)$.
    \soln{9cm}{Identifying $f=5 x + 6$ and $g=\left(5 x + 4\right) \cos{\left(x \right)}$ and using the product rule with $f=5 x + 6 \implies f'=5$. This leaves g as $g = \left(5 x + 4\right) \cos{\left(x \right)}$ 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 $g=5 x + 4 \implies g'=5$. Popping up a level gives $g'=(5 x + 4)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(5)$Popping up again (Back to the original problem) gives $f'=(5 x + 6)(- \left(5 x + 4\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)})+(\left(5 x + 4\right) \cos{\left(x \right)})(5)=- \left(5 x + 4\right) \left(5 x + 6\right) \sin{\left(x \right)} + \left(25 x + 20\right) \cos{\left(x \right)} + \left(25 x + 30\right) \cos{\left(x \right)}$}

\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)})(5 x + 4) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%285%20x%20%2B%206%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%285%20x%20%2B%204%29%20" alt="LaTeX:  \displaystyle y = (5 x + 6)(\cos{\left(x \right)})(5 x + 4) " data-equation-content=" \displaystyle y = (5 x + 6)(\cos{\left(x \right)})(5 x + 4) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=5 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%20%2B%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(5 x + 4\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%285%20x%20%2B%204%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(5 x + 4\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g=\left(5 x + 4\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%3D5%20x%20%2B%206%20%5Cimplies%20f%27%3D5%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(5 x + 4\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%285%20x%20%2B%204%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(5 x + 4\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g = \left(5 x + 4\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=5 x + 4 \implies g'=5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D5%20x%20%2B%204%20%5Cimplies%20g%27%3D5%20" alt="LaTeX:  \displaystyle g=5 x + 4 \implies g'=5 " data-equation-content=" \displaystyle g=5 x + 4 \implies g'=5 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(5 x + 4)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%285%20x%20%2B%204%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%285%29%20" alt="LaTeX:  \displaystyle g'=(5 x + 4)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(5) " data-equation-content=" \displaystyle g'=(5 x + 4)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(5) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(5 x + 6)(- \left(5 x + 4\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)})+(\left(5 x + 4\right) \cos{\left(x \right)})(5)=- \left(5 x + 4\right) \left(5 x + 6\right) \sin{\left(x \right)} + \left(25 x + 20\right) \cos{\left(x \right)} + \left(25 x + 30\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%285%20x%20%2B%206%29%28-%20%5Cleft%285%20x%20%2B%204%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%205%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%285%20x%20%2B%204%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%285%29%3D-%20%5Cleft%285%20x%20%2B%204%5Cright%29%20%5Cleft%285%20x%20%2B%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2825%20x%20%2B%2020%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2825%20x%20%2B%2030%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(5 x + 6)(- \left(5 x + 4\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)})+(\left(5 x + 4\right) \cos{\left(x \right)})(5)=- \left(5 x + 4\right) \left(5 x + 6\right) \sin{\left(x \right)} + \left(25 x + 20\right) \cos{\left(x \right)} + \left(25 x + 30\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(5 x + 6)(- \left(5 x + 4\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)})+(\left(5 x + 4\right) \cos{\left(x \right)})(5)=- \left(5 x + 4\right) \left(5 x + 6\right) \sin{\left(x \right)} + \left(25 x + 20\right) \cos{\left(x \right)} + \left(25 x + 30\right) \cos{\left(x \right)} " /> </p> </p>