\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the derivative of \(\displaystyle y = (5 x^{3} - 4 x^{2} - 3 x + 5)(5 x^{3} + x^{2} - 7 x + 5)(4 x^{3} + 4 x^{2} + 6 x - 6)\).
Identifying \(\displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5\) and \(\displaystyle g=\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right)\) and using the product rule with \(\displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 \implies f'=15 x^{2} - 8 x - 3\). This leaves g as \(\displaystyle g = \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=5 x^{3} + x^{2} - 7 x + 5 \implies f'=15 x^{2} + 2 x - 7\) and \(\displaystyle g=4 x^{3} + 4 x^{2} + 6 x - 6 \implies g'=12 x^{2} + 8 x + 6\). Popping up a level gives \(\displaystyle g'=(4 x^{3} + 4 x^{2} + 6 x - 6)(15 x^{2} + 2 x - 7)+(5 x^{3} + x^{2} - 7 x + 5)(12 x^{2} + 8 x + 6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(5 x^{3} - 4 x^{2} - 3 x + 5)(\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right))+(\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right))(15 x^{2} - 8 x - 3)=\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} - 8 x - 3\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right)\)
\begin{question}Find the derivative of $y = (5 x^{3} - 4 x^{2} - 3 x + 5)(5 x^{3} + x^{2} - 7 x + 5)(4 x^{3} + 4 x^{2} + 6 x - 6)$.
\soln{9cm}{Identifying $f=5 x^{3} - 4 x^{2} - 3 x + 5$ and $g=\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right)$ and using the product rule with $f=5 x^{3} - 4 x^{2} - 3 x + 5 \implies f'=15 x^{2} - 8 x - 3$. This leaves g as $g = \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right)$ which also requires the product rule. Pushing down in the new product rule $f=5 x^{3} + x^{2} - 7 x + 5 \implies f'=15 x^{2} + 2 x - 7$ and $g=4 x^{3} + 4 x^{2} + 6 x - 6 \implies g'=12 x^{2} + 8 x + 6$. Popping up a level gives $g'=(4 x^{3} + 4 x^{2} + 6 x - 6)(15 x^{2} + 2 x - 7)+(5 x^{3} + x^{2} - 7 x + 5)(12 x^{2} + 8 x + 6)$Popping up again (Back to the original problem) gives $f'=(5 x^{3} - 4 x^{2} - 3 x + 5)(\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right))+(\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right))(15 x^{2} - 8 x - 3)=\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} - 8 x - 3\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right)$}
\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^{3} - 4 x^{2} - 3 x + 5)(5 x^{3} + x^{2} - 7 x + 5)(4 x^{3} + 4 x^{2} + 6 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%285%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%203%20x%20%2B%205%29%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%29%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%29%20" alt="LaTeX: \displaystyle y = (5 x^{3} - 4 x^{2} - 3 x + 5)(5 x^{3} + x^{2} - 7 x + 5)(4 x^{3} + 4 x^{2} + 6 x - 6) " data-equation-content=" \displaystyle y = (5 x^{3} - 4 x^{2} - 3 x + 5)(5 x^{3} + x^{2} - 7 x + 5)(4 x^{3} + 4 x^{2} + 6 x - 6) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%203%20x%20%2B%205%20" alt="LaTeX: \displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 " data-equation-content=" \displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 " /> and <img class="equation_image" title=" \displaystyle g=\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) " data-equation-content=" \displaystyle g=\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 \implies f'=15 x^{2} - 8 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%203%20x%20%2B%205%20%5Cimplies%20f%27%3D15%20x%5E%7B2%7D%20-%208%20x%20-%203%20" alt="LaTeX: \displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 \implies f'=15 x^{2} - 8 x - 3 " data-equation-content=" \displaystyle f=5 x^{3} - 4 x^{2} - 3 x + 5 \implies f'=15 x^{2} - 8 x - 3 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) " data-equation-content=" \displaystyle g = \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=5 x^{3} + x^{2} - 7 x + 5 \implies f'=15 x^{2} + 2 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%20%5Cimplies%20f%27%3D15%20x%5E%7B2%7D%20%2B%202%20x%20-%207%20" alt="LaTeX: \displaystyle f=5 x^{3} + x^{2} - 7 x + 5 \implies f'=15 x^{2} + 2 x - 7 " data-equation-content=" \displaystyle f=5 x^{3} + x^{2} - 7 x + 5 \implies f'=15 x^{2} + 2 x - 7 " /> and <img class="equation_image" title=" \displaystyle g=4 x^{3} + 4 x^{2} + 6 x - 6 \implies g'=12 x^{2} + 8 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D4%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%20%5Cimplies%20g%27%3D12%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%20" alt="LaTeX: \displaystyle g=4 x^{3} + 4 x^{2} + 6 x - 6 \implies g'=12 x^{2} + 8 x + 6 " data-equation-content=" \displaystyle g=4 x^{3} + 4 x^{2} + 6 x - 6 \implies g'=12 x^{2} + 8 x + 6 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(4 x^{3} + 4 x^{2} + 6 x - 6)(15 x^{2} + 2 x - 7)+(5 x^{3} + x^{2} - 7 x + 5)(12 x^{2} + 8 x + 6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%29%2815%20x%5E%7B2%7D%20%2B%202%20x%20-%207%29%2B%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%29%2812%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%29%20" alt="LaTeX: \displaystyle g'=(4 x^{3} + 4 x^{2} + 6 x - 6)(15 x^{2} + 2 x - 7)+(5 x^{3} + x^{2} - 7 x + 5)(12 x^{2} + 8 x + 6) " data-equation-content=" \displaystyle g'=(4 x^{3} + 4 x^{2} + 6 x - 6)(15 x^{2} + 2 x - 7)+(5 x^{3} + x^{2} - 7 x + 5)(12 x^{2} + 8 x + 6) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(5 x^{3} - 4 x^{2} - 3 x + 5)(\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right))+(\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right))(15 x^{2} - 8 x - 3)=\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} - 8 x - 3\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%285%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%203%20x%20%2B%205%29%28%5Cleft%2812%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%5Cright%29%20%2B%20%5Cleft%2815%20x%5E%7B2%7D%20%2B%202%20x%20-%207%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%5Cright%29%29%2B%28%5Cleft%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%5Cright%29%29%2815%20x%5E%7B2%7D%20-%208%20x%20-%203%29%3D%5Cleft%2812%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%203%20x%20%2B%205%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%5Cright%29%20%2B%20%5Cleft%2815%20x%5E%7B2%7D%20-%208%20x%20-%203%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%205%5Cright%29%20%2B%20%5Cleft%2815%20x%5E%7B2%7D%20%2B%202%20x%20-%207%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%206%20x%20-%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%203%20x%20%2B%205%5Cright%29%20" alt="LaTeX: \displaystyle f'=(5 x^{3} - 4 x^{2} - 3 x + 5)(\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right))+(\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right))(15 x^{2} - 8 x - 3)=\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} - 8 x - 3\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) " data-equation-content=" \displaystyle f'=(5 x^{3} - 4 x^{2} - 3 x + 5)(\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right))+(\left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right))(15 x^{2} - 8 x - 3)=\left(12 x^{2} + 8 x + 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} - 8 x - 3\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} + x^{2} - 7 x + 5\right) + \left(15 x^{2} + 2 x - 7\right) \left(4 x^{3} + 4 x^{2} + 6 x - 6\right) \left(5 x^{3} - 4 x^{2} - 3 x + 5\right) " /> </p> </p>