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Calculus
Derivatives
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Find the derivative of \(\displaystyle y = e^{x} \cos{\left(x \right)}\).

Using the product rule with \(\displaystyle f = \cos{\left(x \right)}\), \(\displaystyle f' = - \sin{\left(x \right)}\), \(\displaystyle g = e^{x}\), and \(\displaystyle g'= e^{x}\) gives:
\begin{equation*} y' = (e^{x})(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(e^{x}) = - e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = e^{x} \cos{\left(x \right)}$. 
    \soln{9cm}{Using the product rule with $f = \cos{\left(x \right)}$, $f' = - \sin{\left(x \right)}$, $g = e^{x}$, and $g'= e^{x}$ gives:\newline 
 \begin{equation*} y' =  (e^{x})(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(e^{x}) = - e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} \end{equation*}}

\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 = e^{x} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle y = e^{x} \cos{\left(x \right)} " data-equation-content=" \displaystyle y = e^{x} \cos{\left(x \right)} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the product rule with  <img class="equation_image" title=" \displaystyle f = \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%20%3D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle f = \cos{\left(x \right)} " data-equation-content=" \displaystyle f = \cos{\left(x \right)} " /> ,  <img class="equation_image" title=" \displaystyle f' = - \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%20%3D%20-%20%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)} " /> ,  <img class="equation_image" title=" \displaystyle g = e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g = e^{x} " data-equation-content=" \displaystyle g = e^{x} " /> , and  <img class="equation_image" title=" \displaystyle g'= e^{x} " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%20e%5E%7Bx%7D%20" alt="LaTeX:  \displaystyle g'= e^{x} " data-equation-content=" \displaystyle g'= e^{x} " />  gives:<br> 
  <img class="equation_image" title="  y' =  (e^{x})(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(e^{x}) = - e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}  " src="/equation_images/%20%20y%27%20%3D%20%20%28e%5E%7Bx%7D%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20%2B%20%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%20%3D%20-%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' =  (e^{x})(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(e^{x}) = - e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}  " data-equation-content="  y' =  (e^{x})(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(e^{x}) = - e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}  " /> </p> </p>