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Questions: Algebra BusinessCalculus
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Find the derivative of
Using the product rule with
\begin{question}Find the derivative of $y = e^{x} \sin{\left(x \right)}$.
\soln{9cm}{Using the product rule with $f = e^{x}$, $f' = e^{x}$, $g = \sin{\left(x \right)}$, and $g'= \cos{\left(x \right)}$ gives:\newline
\begin{equation*} y' = (\sin{\left(x \right)})(e^{x}) + (e^{x})(\cos{\left(x \right)}) = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} \end{equation*}}
\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} \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle y = e^{x} \sin{\left(x \right)} " data-equation-content=" \displaystyle y = e^{x} \sin{\left(x \right)} " /> . </p> </p><p> <p>Using the product rule with <img class="equation_image" title=" \displaystyle f = e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%20%3D%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f = e^{x} " data-equation-content=" \displaystyle f = e^{x} " /> , <img class="equation_image" title=" \displaystyle f' = e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%20%3D%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f' = e^{x} " data-equation-content=" \displaystyle f' = e^{x} " /> , <img class="equation_image" title=" \displaystyle g = \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \sin{\left(x \right)} " data-equation-content=" \displaystyle g = \sin{\left(x \right)} " /> , and <img class="equation_image" title=" \displaystyle g'= \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g'= \cos{\left(x \right)} " data-equation-content=" \displaystyle g'= \cos{\left(x \right)} " /> gives:<br>
<img class="equation_image" title=" y' = (\sin{\left(x \right)})(e^{x}) + (e^{x})(\cos{\left(x \right)}) = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} " src="/equation_images/%20%20y%27%20%3D%20%20%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%20%2B%20%28e%5E%7Bx%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20%3D%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%20%20" alt="LaTeX: y' = (\sin{\left(x \right)})(e^{x}) + (e^{x})(\cos{\left(x \right)}) = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} " data-equation-content=" y' = (\sin{\left(x \right)})(e^{x}) + (e^{x})(\cos{\left(x \right)}) = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} " /> </p> </p>