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Find the derivative of \(\displaystyle y = (\sin{\left(x \right)})(\cos{\left(x \right)})(7 x^{3} + 4 x^{2} + 7 x + 8)\).
Identifying \(\displaystyle f=\sin{\left(x \right)}\) and \(\displaystyle g=\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)}\) and using the product rule with \(\displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}\). This leaves g as \(\displaystyle g = \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)}\) 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 \(\displaystyle g=7 x^{3} + 4 x^{2} + 7 x + 8 \implies g'=21 x^{2} + 8 x + 7\). Popping up a level gives \(\displaystyle g'=(7 x^{3} + 4 x^{2} + 7 x + 8)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(21 x^{2} + 8 x + 7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(\sin{\left(x \right)})(\left(21 x^{2} + 8 x + 7\right) \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin{\left(x \right)})+(\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)})(\cos{\left(x \right)})=\left(21 x^{2} + 8 x + 7\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin^{2}{\left(x \right)} + \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos^{2}{\left(x \right)}\)
\begin{question}Find the derivative of $y = (\sin{\left(x \right)})(\cos{\left(x \right)})(7 x^{3} + 4 x^{2} + 7 x + 8)$.
\soln{9cm}{Identifying $f=\sin{\left(x \right)}$ and $g=\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)}$ and using the product rule with $f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}$. This leaves g as $g = \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)}$ 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 $g=7 x^{3} + 4 x^{2} + 7 x + 8 \implies g'=21 x^{2} + 8 x + 7$. Popping up a level gives $g'=(7 x^{3} + 4 x^{2} + 7 x + 8)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(21 x^{2} + 8 x + 7)$Popping up again (Back to the original problem) gives $f'=(\sin{\left(x \right)})(\left(21 x^{2} + 8 x + 7\right) \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin{\left(x \right)})+(\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)})(\cos{\left(x \right)})=\left(21 x^{2} + 8 x + 7\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin^{2}{\left(x \right)} + \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos^{2}{\left(x \right)}$}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the derivative of <img class="equation_image" title=" \displaystyle y = (\sin{\left(x \right)})(\cos{\left(x \right)})(7 x^{3} + 4 x^{2} + 7 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle y = (\sin{\left(x \right)})(\cos{\left(x \right)})(7 x^{3} + 4 x^{2} + 7 x + 8) " data-equation-content=" \displaystyle y = (\sin{\left(x \right)})(\cos{\left(x \right)})(7 x^{3} + 4 x^{2} + 7 x + 8) " /> .</p> </p><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(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g=\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \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(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g = \left(7 x^{3} + 4 x^{2} + 7 x + 8\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=7 x^{3} + 4 x^{2} + 7 x + 8 \implies g'=21 x^{2} + 8 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D7%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%20%5Cimplies%20g%27%3D21%20x%5E%7B2%7D%20%2B%208%20x%20%2B%207%20" alt="LaTeX: \displaystyle g=7 x^{3} + 4 x^{2} + 7 x + 8 \implies g'=21 x^{2} + 8 x + 7 " data-equation-content=" \displaystyle g=7 x^{3} + 4 x^{2} + 7 x + 8 \implies g'=21 x^{2} + 8 x + 7 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(7 x^{3} + 4 x^{2} + 7 x + 8)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(21 x^{2} + 8 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2821%20x%5E%7B2%7D%20%2B%208%20x%20%2B%207%29%20" alt="LaTeX: \displaystyle g'=(7 x^{3} + 4 x^{2} + 7 x + 8)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(21 x^{2} + 8 x + 7) " data-equation-content=" \displaystyle g'=(7 x^{3} + 4 x^{2} + 7 x + 8)(- \sin{\left(x \right)})+(\cos{\left(x \right)})(21 x^{2} + 8 x + 7) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(\sin{\left(x \right)})(\left(21 x^{2} + 8 x + 7\right) \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin{\left(x \right)})+(\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)})(\cos{\left(x \right)})=\left(21 x^{2} + 8 x + 7\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin^{2}{\left(x \right)} + \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos^{2}{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Cleft%2821%20x%5E%7B2%7D%20%2B%208%20x%20%2B%207%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%3D%5Cleft%2821%20x%5E%7B2%7D%20%2B%208%20x%20%2B%207%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%5Cright%29%20%5Csin%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%287%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%208%5Cright%29%20%5Ccos%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f'=(\sin{\left(x \right)})(\left(21 x^{2} + 8 x + 7\right) \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin{\left(x \right)})+(\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)})(\cos{\left(x \right)})=\left(21 x^{2} + 8 x + 7\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin^{2}{\left(x \right)} + \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos^{2}{\left(x \right)} " data-equation-content=" \displaystyle f'=(\sin{\left(x \right)})(\left(21 x^{2} + 8 x + 7\right) \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin{\left(x \right)})+(\left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos{\left(x \right)})(\cos{\left(x \right)})=\left(21 x^{2} + 8 x + 7\right) \sin{\left(x \right)} \cos{\left(x \right)} - \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \sin^{2}{\left(x \right)} + \left(7 x^{3} + 4 x^{2} + 7 x + 8\right) \cos^{2}{\left(x \right)} " /> </p> </p>