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

Using the quotient 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*} \frac{ (e^{x})(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(e^{x})}{(e^{x})^2} = - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = e^{- x} \cos{\left(x \right)}$. 
    \soln{9cm}{Using the quotient rule with $f = \cos{\left(x \right)}$, $f' = - \sin{\left(x \right)}$, $g = e^{x}$, and $g'= e^{x}$ gives:\newline 
 \begin{equation*} \frac{ (e^{x})(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(e^{x})}{(e^{x})^2} = - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x} \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%7B-%20x%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 quotient 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="  \frac{ (e^{x})(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(e^{x})}{(e^{x})^2} = - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x}  " src="/equation_images/%20%20%5Cfrac%7B%20%28e%5E%7Bx%7D%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20-%20%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%7D%7B%28e%5E%7Bx%7D%29%5E2%7D%20%3D%20-%20%5Cleft%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%5Cright%29%20e%5E%7B-%20x%7D%20%20" alt="LaTeX:   \frac{ (e^{x})(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(e^{x})}{(e^{x})^2} = - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x}  " data-equation-content="  \frac{ (e^{x})(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(e^{x})}{(e^{x})^2} = - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- x}  " /> </p> </p>