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Calculus
Derivatives
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Find the derivative of \(\displaystyle y = \frac{\sqrt{x}}{- 6 x^{2} + 8 x + 4}\).

Using the quotient rule with \(\displaystyle f = \sqrt{x}\), \(\displaystyle f' = \frac{1}{2 \sqrt{x}}\), \(\displaystyle g = - 6 x^{2} + 8 x + 4\), and \(\displaystyle g'= 8 - 12 x\) gives:
\begin{equation*} \frac{ (- 6 x^{2} + 8 x + 4)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(8 - 12 x)}{(- 6 x^{2} + 8 x + 4)^2} = \frac{9 x^{2} - 4 x + 2}{4 \sqrt{x} \left(3 x^{2} - 4 x - 2\right)^{2}} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \frac{\sqrt{x}}{- 6 x^{2} + 8 x + 4}$. 
    \soln{9cm}{Using the quotient rule with $f = \sqrt{x}$, $f' = \frac{1}{2 \sqrt{x}}$, $g = - 6 x^{2} + 8 x + 4$, and $g'= 8 - 12 x$ gives:\newline 
 \begin{equation*} \frac{ (- 6 x^{2} + 8 x + 4)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(8 - 12 x)}{(- 6 x^{2} + 8 x + 4)^2} = \frac{9 x^{2} - 4 x + 2}{4 \sqrt{x} \left(3 x^{2} - 4 x - 2\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{\sqrt{x}}{- 6 x^{2} + 8 x + 4} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%7D%7B-%206%20x%5E%7B2%7D%20%2B%208%20x%20%2B%204%7D%20" alt="LaTeX:  \displaystyle y = \frac{\sqrt{x}}{- 6 x^{2} + 8 x + 4} " data-equation-content=" \displaystyle y = \frac{\sqrt{x}}{- 6 x^{2} + 8 x + 4} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the quotient rule with  <img class="equation_image" title=" \displaystyle f = \sqrt{x} " src="/equation_images/%20%5Cdisplaystyle%20f%20%3D%20%5Csqrt%7Bx%7D%20" alt="LaTeX:  \displaystyle f = \sqrt{x} " data-equation-content=" \displaystyle f = \sqrt{x} " /> ,  <img class="equation_image" title=" \displaystyle f' = \frac{1}{2 \sqrt{x}} " src="/equation_images/%20%5Cdisplaystyle%20f%27%20%3D%20%5Cfrac%7B1%7D%7B2%20%5Csqrt%7Bx%7D%7D%20" alt="LaTeX:  \displaystyle f' = \frac{1}{2 \sqrt{x}} " data-equation-content=" \displaystyle f' = \frac{1}{2 \sqrt{x}} " /> ,  <img class="equation_image" title=" \displaystyle g = - 6 x^{2} + 8 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20-%206%20x%5E%7B2%7D%20%2B%208%20x%20%2B%204%20" alt="LaTeX:  \displaystyle g = - 6 x^{2} + 8 x + 4 " data-equation-content=" \displaystyle g = - 6 x^{2} + 8 x + 4 " /> , and  <img class="equation_image" title=" \displaystyle g'= 8 - 12 x " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%208%20-%2012%20x%20" alt="LaTeX:  \displaystyle g'= 8 - 12 x " data-equation-content=" \displaystyle g'= 8 - 12 x " />  gives:<br> 
  <img class="equation_image" title="  \frac{ (- 6 x^{2} + 8 x + 4)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(8 - 12 x)}{(- 6 x^{2} + 8 x + 4)^2} = \frac{9 x^{2} - 4 x + 2}{4 \sqrt{x} \left(3 x^{2} - 4 x - 2\right)^{2}}  " src="/equation_images/%20%20%5Cfrac%7B%20%28-%206%20x%5E%7B2%7D%20%2B%208%20x%20%2B%204%29%28%5Cfrac%7B1%7D%7B2%20%5Csqrt%7Bx%7D%7D%29%20-%20%28%5Csqrt%7Bx%7D%29%288%20-%2012%20x%29%7D%7B%28-%206%20x%5E%7B2%7D%20%2B%208%20x%20%2B%204%29%5E2%7D%20%3D%20%5Cfrac%7B9%20x%5E%7B2%7D%20-%204%20x%20%2B%202%7D%7B4%20%5Csqrt%7Bx%7D%20%5Cleft%283%20x%5E%7B2%7D%20-%204%20x%20-%202%5Cright%29%5E%7B2%7D%7D%20%20" alt="LaTeX:   \frac{ (- 6 x^{2} + 8 x + 4)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(8 - 12 x)}{(- 6 x^{2} + 8 x + 4)^2} = \frac{9 x^{2} - 4 x + 2}{4 \sqrt{x} \left(3 x^{2} - 4 x - 2\right)^{2}}  " data-equation-content="  \frac{ (- 6 x^{2} + 8 x + 4)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(8 - 12 x)}{(- 6 x^{2} + 8 x + 4)^2} = \frac{9 x^{2} - 4 x + 2}{4 \sqrt{x} \left(3 x^{2} - 4 x - 2\right)^{2}}  " /> </p> </p>