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

Please login to create an exam or a quiz.

Calculus
Derivatives
New Random

Find the derivative of \(\displaystyle y = \left(x^{2} - 8 x - 5\right) \cos{\left(x \right)}\).

Using the product rule with \(\displaystyle f = \cos{\left(x \right)}\), \(\displaystyle f' = - \sin{\left(x \right)}\), \(\displaystyle g = x^{2} - 8 x - 5\), and \(\displaystyle g'= 2 x - 8\) gives:
\begin{equation*} y' = (x^{2} - 8 x - 5)(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(2 x - 8) = \left(2 x - 8\right) \cos{\left(x \right)} - \left(x^{2} - 8 x - 5\right) \sin{\left(x \right)} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Find the derivative of $y = \left(x^{2} - 8 x - 5\right) \cos{\left(x \right)}$. 
    \soln{9cm}{Using the product rule with $f = \cos{\left(x \right)}$, $f' = - \sin{\left(x \right)}$, $g = x^{2} - 8 x - 5$, and $g'= 2 x - 8$ gives:\newline 
 \begin{equation*} y' =  (x^{2} - 8 x - 5)(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(2 x - 8) = \left(2 x - 8\right) \cos{\left(x \right)} - \left(x^{2} - 8 x - 5\right) \sin{\left(x \right)} \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 = \left(x^{2} - 8 x - 5\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cleft%28x%5E%7B2%7D%20-%208%20x%20-%205%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle y = \left(x^{2} - 8 x - 5\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle y = \left(x^{2} - 8 x - 5\right) \cos{\left(x \right)} " /> . </p> </p>
HTML for Canvas
<p> <p>Using the product 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 = x^{2} - 8 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20x%5E%7B2%7D%20-%208%20x%20-%205%20" alt="LaTeX:  \displaystyle g = x^{2} - 8 x - 5 " data-equation-content=" \displaystyle g = x^{2} - 8 x - 5 " /> , and  <img class="equation_image" title=" \displaystyle g'= 2 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%202%20x%20-%208%20" alt="LaTeX:  \displaystyle g'= 2 x - 8 " data-equation-content=" \displaystyle g'= 2 x - 8 " />  gives:<br> 
  <img class="equation_image" title="  y' =  (x^{2} - 8 x - 5)(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(2 x - 8) = \left(2 x - 8\right) \cos{\left(x \right)} - \left(x^{2} - 8 x - 5\right) \sin{\left(x \right)}  " src="/equation_images/%20%20y%27%20%3D%20%20%28x%5E%7B2%7D%20-%208%20x%20-%205%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20%2B%20%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%282%20x%20-%208%29%20%3D%20%5Cleft%282%20x%20-%208%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%28x%5E%7B2%7D%20-%208%20x%20-%205%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%20" alt="LaTeX:   y' =  (x^{2} - 8 x - 5)(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(2 x - 8) = \left(2 x - 8\right) \cos{\left(x \right)} - \left(x^{2} - 8 x - 5\right) \sin{\left(x \right)}  " data-equation-content="  y' =  (x^{2} - 8 x - 5)(- \sin{\left(x \right)}) + (\cos{\left(x \right)})(2 x - 8) = \left(2 x - 8\right) \cos{\left(x \right)} - \left(x^{2} - 8 x - 5\right) \sin{\left(x \right)}  " /> </p> </p>