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Calculus
Derivatives
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Find the derivative of \(\displaystyle y = \frac{4 x^{2} + 4 x + 7}{\sin{\left(x \right)}}\).

Using the quotient rule with \(\displaystyle f = 4 x^{2} + 4 x + 7\), \(\displaystyle f' = 8 x + 4\), \(\displaystyle g = \sin{\left(x \right)}\), and \(\displaystyle g'= \cos{\left(x \right)}\) gives:
\begin{equation*} \frac{ (\sin{\left(x \right)})(8 x + 4) - (4 x^{2} + 4 x + 7)(\cos{\left(x \right)})}{(\sin{\left(x \right)})^2} = - \frac{4 x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \frac{4 x^{2} + 4 x + 7}{\sin{\left(x \right)}}$. 
    \soln{9cm}{Using the quotient rule with $f = 4 x^{2} + 4 x + 7$, $f' = 8 x + 4$, $g = \sin{\left(x \right)}$, and $g'= \cos{\left(x \right)}$ gives:\newline 
 \begin{equation*} \frac{ (\sin{\left(x \right)})(8 x + 4) - (4 x^{2} + 4 x + 7)(\cos{\left(x \right)})}{(\sin{\left(x \right)})^2} = - \frac{4 x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\sin^{2}{\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 = \frac{4 x^{2} + 4 x + 7}{\sin{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B4%20x%5E%7B2%7D%20%2B%204%20x%20%2B%207%7D%7B%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y = \frac{4 x^{2} + 4 x + 7}{\sin{\left(x \right)}} " data-equation-content=" \displaystyle y = \frac{4 x^{2} + 4 x + 7}{\sin{\left(x \right)}} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the quotient rule with  <img class="equation_image" title=" \displaystyle f = 4 x^{2} + 4 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20f%20%3D%204%20x%5E%7B2%7D%20%2B%204%20x%20%2B%207%20" alt="LaTeX:  \displaystyle f = 4 x^{2} + 4 x + 7 " data-equation-content=" \displaystyle f = 4 x^{2} + 4 x + 7 " /> ,  <img class="equation_image" title=" \displaystyle f' = 8 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20f%27%20%3D%208%20x%20%2B%204%20" alt="LaTeX:  \displaystyle f' = 8 x + 4 " data-equation-content=" \displaystyle f' = 8 x + 4 " /> ,  <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="  \frac{ (\sin{\left(x \right)})(8 x + 4) - (4 x^{2} + 4 x + 7)(\cos{\left(x \right)})}{(\sin{\left(x \right)})^2} = - \frac{4 x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}  " src="/equation_images/%20%20%5Cfrac%7B%20%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%288%20x%20%2B%204%29%20-%20%284%20x%5E%7B2%7D%20%2B%204%20x%20%2B%207%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%7D%7B%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%5E2%7D%20%3D%20-%20%5Cfrac%7B4%20x%5E%7B2%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%208%20x%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%204%20x%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%204%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%207%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csin%5E%7B2%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%20" alt="LaTeX:   \frac{ (\sin{\left(x \right)})(8 x + 4) - (4 x^{2} + 4 x + 7)(\cos{\left(x \right)})}{(\sin{\left(x \right)})^2} = - \frac{4 x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}  " data-equation-content="  \frac{ (\sin{\left(x \right)})(8 x + 4) - (4 x^{2} + 4 x + 7)(\cos{\left(x \right)})}{(\sin{\left(x \right)})^2} = - \frac{4 x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}  " /> </p> </p>