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

Download \(\LaTeX\)

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

\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.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (\sin{\left(x \right)})(e^{x})(\cos{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle y = (\sin{\left(x \right)})(e^{x})(\cos{\left(x \right)}) " data-equation-content=" \displaystyle y = (\sin{\left(x \right)})(e^{x})(\cos{\left(x \right)}) " /> .</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=e^{x} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3De%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=e^{x} \cos{\left(x \right)} " data-equation-content=" \displaystyle g=e^{x} \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 = e^{x} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = e^{x} \cos{\left(x \right)} " data-equation-content=" \displaystyle g = e^{x} \cos{\left(x \right)} " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=e^{x} \implies f'=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20%5Cimplies%20f%27%3De%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle f=e^{x} \implies f'=e^{x} " data-equation-content=" \displaystyle f=e^{x} \implies f'=e^{x} " />  and  <img class="equation_image" title=" \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(\cos{\left(x \right)})(e^{x})+(e^{x})(- \sin{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%2B%28e%5E%7Bx%7D%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle g'=(\cos{\left(x \right)})(e^{x})+(e^{x})(- \sin{\left(x \right)}) " data-equation-content=" \displaystyle g'=(\cos{\left(x \right)})(e^{x})+(e^{x})(- \sin{\left(x \right)}) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(\sin{\left(x \right)})(- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \cos{\left(x \right)})(\cos{\left(x \right)})=- e^{x} \sin^{2}{\left(x \right)} + e^{x} \sin{\left(x \right)} \cos{\left(x \right)} + e^{x} \cos^{2}{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%3D-%20e%5E%7Bx%7D%20%5Csin%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20e%5E%7Bx%7D%20%5Ccos%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f'=(\sin{\left(x \right)})(- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \cos{\left(x \right)})(\cos{\left(x \right)})=- e^{x} \sin^{2}{\left(x \right)} + e^{x} \sin{\left(x \right)} \cos{\left(x \right)} + e^{x} \cos^{2}{\left(x \right)} " data-equation-content=" \displaystyle f'=(\sin{\left(x \right)})(- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \cos{\left(x \right)})(\cos{\left(x \right)})=- e^{x} \sin^{2}{\left(x \right)} + e^{x} \sin{\left(x \right)} \cos{\left(x \right)} + e^{x} \cos^{2}{\left(x \right)} " /> </p> </p>