\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus

Please login to create an exam or a quiz.

Calculus
Derivatives
New Random

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

Using the quotient rule with \(\displaystyle f = \sqrt{x}\), \(\displaystyle f' = \frac{1}{2 \sqrt{x}}\), \(\displaystyle g = - 8 x^{2} + x - 8\), and \(\displaystyle g'= 1 - 16 x\) gives:
\begin{equation*} \frac{ (- 8 x^{2} + x - 8)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(1 - 16 x)}{(- 8 x^{2} + x - 8)^2} = \frac{24 x^{2} - x - 8}{2 \sqrt{x} \left(8 x^{2} - x + 8\right)^{2}} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \frac{\sqrt{x}}{- 8 x^{2} + x - 8}$. 
    \soln{9cm}{Using the quotient rule with $f = \sqrt{x}$, $f' = \frac{1}{2 \sqrt{x}}$, $g = - 8 x^{2} + x - 8$, and $g'= 1 - 16 x$ gives:\newline 
 \begin{equation*} \frac{ (- 8 x^{2} + x - 8)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(1 - 16 x)}{(- 8 x^{2} + x - 8)^2} = \frac{24 x^{2} - x - 8}{2 \sqrt{x} \left(8 x^{2} - x + 8\right)^{2}} \end{equation*}}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}
HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = \frac{\sqrt{x}}{- 8 x^{2} + x - 8} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%7D%7B-%208%20x%5E%7B2%7D%20%2B%20x%20-%208%7D%20" alt="LaTeX:  \displaystyle y = \frac{\sqrt{x}}{- 8 x^{2} + x - 8} " data-equation-content=" \displaystyle y = \frac{\sqrt{x}}{- 8 x^{2} + x - 8} " /> . </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 = - 8 x^{2} + x - 8 " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20-%208%20x%5E%7B2%7D%20%2B%20x%20-%208%20" alt="LaTeX:  \displaystyle g = - 8 x^{2} + x - 8 " data-equation-content=" \displaystyle g = - 8 x^{2} + x - 8 " /> , and  <img class="equation_image" title=" \displaystyle g'= 1 - 16 x " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%201%20-%2016%20x%20" alt="LaTeX:  \displaystyle g'= 1 - 16 x " data-equation-content=" \displaystyle g'= 1 - 16 x " />  gives:<br> 
  <img class="equation_image" title="  \frac{ (- 8 x^{2} + x - 8)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(1 - 16 x)}{(- 8 x^{2} + x - 8)^2} = \frac{24 x^{2} - x - 8}{2 \sqrt{x} \left(8 x^{2} - x + 8\right)^{2}}  " src="/equation_images/%20%20%5Cfrac%7B%20%28-%208%20x%5E%7B2%7D%20%2B%20x%20-%208%29%28%5Cfrac%7B1%7D%7B2%20%5Csqrt%7Bx%7D%7D%29%20-%20%28%5Csqrt%7Bx%7D%29%281%20-%2016%20x%29%7D%7B%28-%208%20x%5E%7B2%7D%20%2B%20x%20-%208%29%5E2%7D%20%3D%20%5Cfrac%7B24%20x%5E%7B2%7D%20-%20x%20-%208%7D%7B2%20%5Csqrt%7Bx%7D%20%5Cleft%288%20x%5E%7B2%7D%20-%20x%20%2B%208%5Cright%29%5E%7B2%7D%7D%20%20" alt="LaTeX:   \frac{ (- 8 x^{2} + x - 8)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(1 - 16 x)}{(- 8 x^{2} + x - 8)^2} = \frac{24 x^{2} - x - 8}{2 \sqrt{x} \left(8 x^{2} - x + 8\right)^{2}}  " data-equation-content="  \frac{ (- 8 x^{2} + x - 8)(\frac{1}{2 \sqrt{x}}) - (\sqrt{x})(1 - 16 x)}{(- 8 x^{2} + x - 8)^2} = \frac{24 x^{2} - x - 8}{2 \sqrt{x} \left(8 x^{2} - x + 8\right)^{2}}  " /> </p> </p>