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

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

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \frac{\cos{\left(x \right)}}{2 x^{2} - 7 x + 4}$. 
    \soln{9cm}{Using the quotient rule with $f = \cos{\left(x \right)}$, $f' = - \sin{\left(x \right)}$, $g = 2 x^{2} - 7 x + 4$, and $g'= 4 x - 7$ gives:\newline 
 \begin{equation*} \frac{ (2 x^{2} - 7 x + 4)(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(4 x - 7)}{(2 x^{2} - 7 x + 4)^2} = - \frac{2 x^{2} \sin{\left(x \right)} - 7 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 4 \sin{\left(x \right)} - 7 \cos{\left(x \right)}}{\left(2 x^{2} - 7 x + 4\right)^{2}} \end{equation*}}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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\begin{document}\begin{question}(10pts) The question goes here!
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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = \frac{\cos{\left(x \right)}}{2 x^{2} - 7 x + 4} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B2%20x%5E%7B2%7D%20-%207%20x%20%2B%204%7D%20" alt="LaTeX:  \displaystyle y = \frac{\cos{\left(x \right)}}{2 x^{2} - 7 x + 4} " data-equation-content=" \displaystyle y = \frac{\cos{\left(x \right)}}{2 x^{2} - 7 x + 4} " /> . </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 = 2 x^{2} - 7 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%202%20x%5E%7B2%7D%20-%207%20x%20%2B%204%20" alt="LaTeX:  \displaystyle g = 2 x^{2} - 7 x + 4 " data-equation-content=" \displaystyle g = 2 x^{2} - 7 x + 4 " /> , and  <img class="equation_image" title=" \displaystyle g'= 4 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%204%20x%20-%207%20" alt="LaTeX:  \displaystyle g'= 4 x - 7 " data-equation-content=" \displaystyle g'= 4 x - 7 " />  gives:<br> 
  <img class="equation_image" title="  \frac{ (2 x^{2} - 7 x + 4)(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(4 x - 7)}{(2 x^{2} - 7 x + 4)^2} = - \frac{2 x^{2} \sin{\left(x \right)} - 7 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 4 \sin{\left(x \right)} - 7 \cos{\left(x \right)}}{\left(2 x^{2} - 7 x + 4\right)^{2}}  " src="/equation_images/%20%20%5Cfrac%7B%20%282%20x%5E%7B2%7D%20-%207%20x%20%2B%204%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20-%20%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%284%20x%20-%207%29%7D%7B%282%20x%5E%7B2%7D%20-%207%20x%20%2B%204%29%5E2%7D%20%3D%20-%20%5Cfrac%7B2%20x%5E%7B2%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%207%20x%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%204%20x%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%204%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%207%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Cleft%282%20x%5E%7B2%7D%20-%207%20x%20%2B%204%5Cright%29%5E%7B2%7D%7D%20%20" alt="LaTeX:   \frac{ (2 x^{2} - 7 x + 4)(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(4 x - 7)}{(2 x^{2} - 7 x + 4)^2} = - \frac{2 x^{2} \sin{\left(x \right)} - 7 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 4 \sin{\left(x \right)} - 7 \cos{\left(x \right)}}{\left(2 x^{2} - 7 x + 4\right)^{2}}  " data-equation-content="  \frac{ (2 x^{2} - 7 x + 4)(- \sin{\left(x \right)}) - (\cos{\left(x \right)})(4 x - 7)}{(2 x^{2} - 7 x + 4)^2} = - \frac{2 x^{2} \sin{\left(x \right)} - 7 x \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 4 \sin{\left(x \right)} - 7 \cos{\left(x \right)}}{\left(2 x^{2} - 7 x + 4\right)^{2}}  " /> </p> </p>