\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the derivative of \(\displaystyle y = (- 5 x^{2} - 3 x + 9)(8 x^{2} + x + 2)(2 x^{2} + 3 x - 2)\).
Identifying \(\displaystyle f=- 5 x^{2} - 3 x + 9\) and \(\displaystyle g=\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right)\) and using the product rule with \(\displaystyle f=- 5 x^{2} - 3 x + 9 \implies f'=- 10 x - 3\). This leaves g as \(\displaystyle g = \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=8 x^{2} + x + 2 \implies f'=16 x + 1\) and \(\displaystyle g=2 x^{2} + 3 x - 2 \implies g'=4 x + 3\). Popping up a level gives \(\displaystyle g'=(2 x^{2} + 3 x - 2)(16 x + 1)+(8 x^{2} + x + 2)(4 x + 3)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 5 x^{2} - 3 x + 9)(\left(4 x + 3\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(2 x^{2} + 3 x - 2\right))+(\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) + \left(4 x + 3\right) \left(- 5 x^{2} - 3 x + 9\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(- 5 x^{2} - 3 x + 9\right) \left(2 x^{2} + 3 x - 2\right)\)
\begin{question}Find the derivative of $y = (- 5 x^{2} - 3 x + 9)(8 x^{2} + x + 2)(2 x^{2} + 3 x - 2)$.
\soln{9cm}{Identifying $f=- 5 x^{2} - 3 x + 9$ and $g=\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right)$ and using the product rule with $f=- 5 x^{2} - 3 x + 9 \implies f'=- 10 x - 3$. This leaves g as $g = \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right)$ which also requires the product rule. Pushing down in the new product rule $f=8 x^{2} + x + 2 \implies f'=16 x + 1$ and $g=2 x^{2} + 3 x - 2 \implies g'=4 x + 3$. Popping up a level gives $g'=(2 x^{2} + 3 x - 2)(16 x + 1)+(8 x^{2} + x + 2)(4 x + 3)$Popping up again (Back to the original problem) gives $f'=(- 5 x^{2} - 3 x + 9)(\left(4 x + 3\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(2 x^{2} + 3 x - 2\right))+(\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) + \left(4 x + 3\right) \left(- 5 x^{2} - 3 x + 9\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(- 5 x^{2} - 3 x + 9\right) \left(2 x^{2} + 3 x - 2\right)$}
\end{question}
\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}<p> <p>Find the derivative of <img class="equation_image" title=" \displaystyle y = (- 5 x^{2} - 3 x + 9)(8 x^{2} + x + 2)(2 x^{2} + 3 x - 2) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%205%20x%5E%7B2%7D%20-%203%20x%20%2B%209%29%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%29%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%29%20" alt="LaTeX: \displaystyle y = (- 5 x^{2} - 3 x + 9)(8 x^{2} + x + 2)(2 x^{2} + 3 x - 2) " data-equation-content=" \displaystyle y = (- 5 x^{2} - 3 x + 9)(8 x^{2} + x + 2)(2 x^{2} + 3 x - 2) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 5 x^{2} - 3 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B2%7D%20-%203%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=- 5 x^{2} - 3 x + 9 " data-equation-content=" \displaystyle f=- 5 x^{2} - 3 x + 9 " /> and <img class="equation_image" title=" \displaystyle g=\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) " data-equation-content=" \displaystyle g=\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 5 x^{2} - 3 x + 9 \implies f'=- 10 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%205%20x%5E%7B2%7D%20-%203%20x%20%2B%209%20%5Cimplies%20f%27%3D-%2010%20x%20-%203%20" alt="LaTeX: \displaystyle f=- 5 x^{2} - 3 x + 9 \implies f'=- 10 x - 3 " data-equation-content=" \displaystyle f=- 5 x^{2} - 3 x + 9 \implies f'=- 10 x - 3 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) " data-equation-content=" \displaystyle g = \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=8 x^{2} + x + 2 \implies f'=16 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B2%7D%20%2B%20x%20%2B%202%20%5Cimplies%20f%27%3D16%20x%20%2B%201%20" alt="LaTeX: \displaystyle f=8 x^{2} + x + 2 \implies f'=16 x + 1 " data-equation-content=" \displaystyle f=8 x^{2} + x + 2 \implies f'=16 x + 1 " /> and <img class="equation_image" title=" \displaystyle g=2 x^{2} + 3 x - 2 \implies g'=4 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D2%20x%5E%7B2%7D%20%2B%203%20x%20-%202%20%5Cimplies%20g%27%3D4%20x%20%2B%203%20" alt="LaTeX: \displaystyle g=2 x^{2} + 3 x - 2 \implies g'=4 x + 3 " data-equation-content=" \displaystyle g=2 x^{2} + 3 x - 2 \implies g'=4 x + 3 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(2 x^{2} + 3 x - 2)(16 x + 1)+(8 x^{2} + x + 2)(4 x + 3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%29%2816%20x%20%2B%201%29%2B%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%29%284%20x%20%2B%203%29%20" alt="LaTeX: \displaystyle g'=(2 x^{2} + 3 x - 2)(16 x + 1)+(8 x^{2} + x + 2)(4 x + 3) " data-equation-content=" \displaystyle g'=(2 x^{2} + 3 x - 2)(16 x + 1)+(8 x^{2} + x + 2)(4 x + 3) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 5 x^{2} - 3 x + 9)(\left(4 x + 3\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(2 x^{2} + 3 x - 2\right))+(\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) + \left(4 x + 3\right) \left(- 5 x^{2} - 3 x + 9\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(- 5 x^{2} - 3 x + 9\right) \left(2 x^{2} + 3 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%205%20x%5E%7B2%7D%20-%203%20x%20%2B%209%29%28%5Cleft%284%20x%20%2B%203%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%5Cright%29%20%2B%20%5Cleft%2816%20x%20%2B%201%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%29%2B%28%5Cleft%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%5Cright%29%29%28-%2010%20x%20-%203%29%3D%5Cleft%28-%2010%20x%20-%203%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%5Cright%29%20%2B%20%5Cleft%284%20x%20%2B%203%5Cright%29%20%5Cleft%28-%205%20x%5E%7B2%7D%20-%203%20x%20%2B%209%5Cright%29%20%5Cleft%288%20x%5E%7B2%7D%20%2B%20x%20%2B%202%5Cright%29%20%2B%20%5Cleft%2816%20x%20%2B%201%5Cright%29%20%5Cleft%28-%205%20x%5E%7B2%7D%20-%203%20x%20%2B%209%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%20x%20-%202%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 5 x^{2} - 3 x + 9)(\left(4 x + 3\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(2 x^{2} + 3 x - 2\right))+(\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) + \left(4 x + 3\right) \left(- 5 x^{2} - 3 x + 9\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(- 5 x^{2} - 3 x + 9\right) \left(2 x^{2} + 3 x - 2\right) " data-equation-content=" \displaystyle f'=(- 5 x^{2} - 3 x + 9)(\left(4 x + 3\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(2 x^{2} + 3 x - 2\right))+(\left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right))(- 10 x - 3)=\left(- 10 x - 3\right) \left(2 x^{2} + 3 x - 2\right) \left(8 x^{2} + x + 2\right) + \left(4 x + 3\right) \left(- 5 x^{2} - 3 x + 9\right) \left(8 x^{2} + x + 2\right) + \left(16 x + 1\right) \left(- 5 x^{2} - 3 x + 9\right) \left(2 x^{2} + 3 x - 2\right) " /> </p> </p>