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Questions: Algebra BusinessCalculus

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Calculus
Derivatives
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Find the derivative of \(\displaystyle y = \frac{e^{x}}{7 - 2 x}\).

Using the quotient rule with \(\displaystyle f = e^{x}\), \(\displaystyle f' = e^{x}\), \(\displaystyle g = 7 - 2 x\), and \(\displaystyle g'= -2\) gives:
\begin{equation*} \frac{ (7 - 2 x)(e^{x}) - (e^{x})(-2)}{(7 - 2 x)^2} = - \frac{\left(2 x - 9\right) e^{x}}{\left(2 x - 7\right)^{2}} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \frac{e^{x}}{7 - 2 x}$. 
    \soln{9cm}{Using the quotient rule with $f = e^{x}$, $f' = e^{x}$, $g = 7 - 2 x$, and $g'= -2$ gives:\newline 
 \begin{equation*} \frac{ (7 - 2 x)(e^{x}) - (e^{x})(-2)}{(7 - 2 x)^2} = - \frac{\left(2 x - 9\right) e^{x}}{\left(2 x - 7\right)^{2}} \end{equation*}}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = \frac{e^{x}}{7 - 2 x} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7Be%5E%7Bx%7D%7D%7B7%20-%202%20x%7D%20" alt="LaTeX:  \displaystyle y = \frac{e^{x}}{7 - 2 x} " data-equation-content=" \displaystyle y = \frac{e^{x}}{7 - 2 x} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the quotient 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 = 7 - 2 x " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%207%20-%202%20x%20" alt="LaTeX:  \displaystyle g = 7 - 2 x " data-equation-content=" \displaystyle g = 7 - 2 x " /> , and  <img class="equation_image" title=" \displaystyle g'= -2 " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%20-2%20" alt="LaTeX:  \displaystyle g'= -2 " data-equation-content=" \displaystyle g'= -2 " />  gives:<br> 
  <img class="equation_image" title="  \frac{ (7 - 2 x)(e^{x}) - (e^{x})(-2)}{(7 - 2 x)^2} = - \frac{\left(2 x - 9\right) e^{x}}{\left(2 x - 7\right)^{2}}  " src="/equation_images/%20%20%5Cfrac%7B%20%287%20-%202%20x%29%28e%5E%7Bx%7D%29%20-%20%28e%5E%7Bx%7D%29%28-2%29%7D%7B%287%20-%202%20x%29%5E2%7D%20%3D%20-%20%5Cfrac%7B%5Cleft%282%20x%20-%209%5Cright%29%20e%5E%7Bx%7D%7D%7B%5Cleft%282%20x%20-%207%5Cright%29%5E%7B2%7D%7D%20%20" alt="LaTeX:   \frac{ (7 - 2 x)(e^{x}) - (e^{x})(-2)}{(7 - 2 x)^2} = - \frac{\left(2 x - 9\right) e^{x}}{\left(2 x - 7\right)^{2}}  " data-equation-content="  \frac{ (7 - 2 x)(e^{x}) - (e^{x})(-2)}{(7 - 2 x)^2} = - \frac{\left(2 x - 9\right) e^{x}}{\left(2 x - 7\right)^{2}}  " /> </p> </p>