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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (e^{x})(\cos{\left(x \right)})(\log{\left(x \right)})\).
Identifying \(\displaystyle f=e^{x}\) and \(\displaystyle g=\log{\left(x \right)} \cos{\left(x \right)}\) and using the product rule with \(\displaystyle f=e^{x} \implies f'=e^{x}\). This leaves g as \(\displaystyle g = \log{\left(x \right)} \cos{\left(x \right)}\) 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 \(\displaystyle g=\log{\left(x \right)} \implies g'=\frac{1}{x}\). Popping up a level gives \(\displaystyle g'=(\log{\left(x \right)})(- \sin{\left(x \right)})+(\cos{\left(x \right)})(\frac{1}{x})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(e^{x})(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x})+(\log{\left(x \right)} \cos{\left(x \right)})(e^{x})=- e^{x} \log{\left(x \right)} \sin{\left(x \right)} + e^{x} \log{\left(x \right)} \cos{\left(x \right)} + \frac{e^{x} \cos{\left(x \right)}}{x}\)
\begin{question}Find the derivative of $y = (e^{x})(\cos{\left(x \right)})(\log{\left(x \right)})$.
\soln{9cm}{Identifying $f=e^{x}$ and $g=\log{\left(x \right)} \cos{\left(x \right)}$ and using the product rule with $f=e^{x} \implies f'=e^{x}$. This leaves g as $g = \log{\left(x \right)} \cos{\left(x \right)}$ 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 $g=\log{\left(x \right)} \implies g'=\frac{1}{x}$. Popping up a level gives $g'=(\log{\left(x \right)})(- \sin{\left(x \right)})+(\cos{\left(x \right)})(\frac{1}{x})$Popping up again (Back to the original problem) gives $f'=(e^{x})(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x})+(\log{\left(x \right)} \cos{\left(x \right)})(e^{x})=- e^{x} \log{\left(x \right)} \sin{\left(x \right)} + e^{x} \log{\left(x \right)} \cos{\left(x \right)} + \frac{e^{x} \cos{\left(x \right)}}{x}$}
\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 = (e^{x})(\cos{\left(x \right)})(\log{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28e%5E%7Bx%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX: \displaystyle y = (e^{x})(\cos{\left(x \right)})(\log{\left(x \right)}) " data-equation-content=" \displaystyle y = (e^{x})(\cos{\left(x \right)})(\log{\left(x \right)}) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f=e^{x} " data-equation-content=" \displaystyle f=e^{x} " /> and <img class="equation_image" title=" \displaystyle g=\log{\left(x \right)} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\log{\left(x \right)} \cos{\left(x \right)} " data-equation-content=" \displaystyle g=\log{\left(x \right)} \cos{\left(x \right)} " /> and using the product rule with <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} " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \log{\left(x \right)} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \log{\left(x \right)} \cos{\left(x \right)} " data-equation-content=" \displaystyle g = \log{\left(x \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=\log{\left(x \right)} \implies g'=\frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX: \displaystyle g=\log{\left(x \right)} \implies g'=\frac{1}{x} " data-equation-content=" \displaystyle g=\log{\left(x \right)} \implies g'=\frac{1}{x} " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(\log{\left(x \right)})(- \sin{\left(x \right)})+(\cos{\left(x \right)})(\frac{1}{x}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Cfrac%7B1%7D%7Bx%7D%29%20" alt="LaTeX: \displaystyle g'=(\log{\left(x \right)})(- \sin{\left(x \right)})+(\cos{\left(x \right)})(\frac{1}{x}) " data-equation-content=" \displaystyle g'=(\log{\left(x \right)})(- \sin{\left(x \right)})+(\cos{\left(x \right)})(\frac{1}{x}) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(e^{x})(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x})+(\log{\left(x \right)} \cos{\left(x \right)})(e^{x})=- e^{x} \log{\left(x \right)} \sin{\left(x \right)} + e^{x} \log{\left(x \right)} \cos{\left(x \right)} + \frac{e^{x} \cos{\left(x \right)}}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28e%5E%7Bx%7D%29%28-%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7Bx%7D%29%2B%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%3D-%20e%5E%7Bx%7D%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20e%5E%7Bx%7D%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7Be%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7Bx%7D%20" alt="LaTeX: \displaystyle f'=(e^{x})(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x})+(\log{\left(x \right)} \cos{\left(x \right)})(e^{x})=- e^{x} \log{\left(x \right)} \sin{\left(x \right)} + e^{x} \log{\left(x \right)} \cos{\left(x \right)} + \frac{e^{x} \cos{\left(x \right)}}{x} " data-equation-content=" \displaystyle f'=(e^{x})(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x})+(\log{\left(x \right)} \cos{\left(x \right)})(e^{x})=- e^{x} \log{\left(x \right)} \sin{\left(x \right)} + e^{x} \log{\left(x \right)} \cos{\left(x \right)} + \frac{e^{x} \cos{\left(x \right)}}{x} " /> </p> </p>