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Find the derivative of \(\displaystyle y = \frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}}\)
Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: \begin{equation*}\ln(y) = \ln{\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)} \end{equation*} Expanding the right hand side using the product and quotient properties of logarithms gives: \begin{equation*}\ln(y) = x + 5 \ln{\left(5 x + 6 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(x \right)} - 8 \ln{\left(5 x - 2 \right)} - 2 \ln{\left(7 \right)} \end{equation*} Taking the derivative on both sides of the equation yields: \begin{equation*}\frac{y'}{y} = - \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x} \end{equation*} Solving for \(\displaystyle y'\) and substituting out y using the original equation gives \begin{equation*}y' = \left(- \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) \end{equation*} Using some Trigonometric identities to simplify gives \begin{equation*}y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{25}{5 x + 6}- \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) \end{equation*}
\begin{question}Find the derivative of $y = \frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}}$ \soln{9cm}{Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: \begin{equation*}\ln(y) = \ln{\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)} \end{equation*} Expanding the right hand side using the product and quotient properties of logarithms gives: \begin{equation*}\ln(y) = x + 5 \ln{\left(5 x + 6 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(x \right)} - 8 \ln{\left(5 x - 2 \right)} - 2 \ln{\left(7 \right)} \end{equation*} Taking the derivative on both sides of the equation yields: \begin{equation*}\frac{y'}{y} = - \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x} \end{equation*} Solving for $y'$ and substituting out y using the original equation gives \begin{equation*}y' = \left(- \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) \end{equation*} Using some Trigonometric identities to simplify gives \begin{equation*}y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{25}{5 x + 6}- \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) \end{equation*} } \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 = \frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B%5Cleft%285%20x%20%2B%206%5Cright%29%5E%7B5%7D%20e%5E%7Bx%7D%20%5Ccos%5E%7B7%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B49%20x%5E%7B2%7D%20%5Cleft%285%20x%20-%202%5Cright%29%5E%7B8%7D%7D%20" alt="LaTeX: \displaystyle y = \frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} " data-equation-content=" \displaystyle y = \frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} " /> </p> </p>
<p> <p>Taking the natural logarithm of both sides of the equation and expanding the right hand side gives:
<img class="equation_image" title=" \ln(y) = \ln{\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)} " src="/equation_images/%20%5Cln%28y%29%20%3D%20%5Cln%7B%5Cleft%28%5Cfrac%7B%5Cleft%285%20x%20%2B%206%5Cright%29%5E%7B5%7D%20e%5E%7Bx%7D%20%5Ccos%5E%7B7%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B49%20x%5E%7B2%7D%20%5Cleft%285%20x%20-%202%5Cright%29%5E%7B8%7D%7D%20%5Cright%29%7D%20%20%20" alt="LaTeX: \ln(y) = \ln{\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)} " data-equation-content=" \ln(y) = \ln{\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)} " />
Expanding the right hand side using the product and quotient properties of logarithms gives:
<img class="equation_image" title=" \ln(y) = x + 5 \ln{\left(5 x + 6 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(x \right)} - 8 \ln{\left(5 x - 2 \right)} - 2 \ln{\left(7 \right)} " src="/equation_images/%20%5Cln%28y%29%20%3D%20x%20%2B%205%20%5Cln%7B%5Cleft%285%20x%20%2B%206%20%5Cright%29%7D%20%2B%207%20%5Cln%7B%5Cleft%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cright%29%7D-%202%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20-%208%20%5Cln%7B%5Cleft%285%20x%20-%202%20%5Cright%29%7D%20-%202%20%5Cln%7B%5Cleft%287%20%5Cright%29%7D%20%20%20" alt="LaTeX: \ln(y) = x + 5 \ln{\left(5 x + 6 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(x \right)} - 8 \ln{\left(5 x - 2 \right)} - 2 \ln{\left(7 \right)} " data-equation-content=" \ln(y) = x + 5 \ln{\left(5 x + 6 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(x \right)} - 8 \ln{\left(5 x - 2 \right)} - 2 \ln{\left(7 \right)} " />
Taking the derivative on both sides of the equation yields:
<img class="equation_image" title=" \frac{y'}{y} = - \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x} " src="/equation_images/%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20-%20%5Cfrac%7B7%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%2B%201%20%2B%20%5Cfrac%7B25%7D%7B5%20x%20%2B%206%7D%20-%20%5Cfrac%7B40%7D%7B5%20x%20-%202%7D%20-%20%5Cfrac%7B2%7D%7Bx%7D%20%20%20" alt="LaTeX: \frac{y'}{y} = - \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x} " data-equation-content=" \frac{y'}{y} = - \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x} " />
Solving for <img class="equation_image" title=" \displaystyle y' " src="/equation_images/%20%5Cdisplaystyle%20y%27%20" alt="LaTeX: \displaystyle y' " data-equation-content=" \displaystyle y' " /> and substituting out y using the original equation gives
<img class="equation_image" title=" y' = \left(- \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) " src="/equation_images/%20y%27%20%3D%20%5Cleft%28-%20%5Cfrac%7B7%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%2B%201%20%2B%20%5Cfrac%7B25%7D%7B5%20x%20%2B%206%7D%20-%20%5Cfrac%7B40%7D%7B5%20x%20-%202%7D%20-%20%5Cfrac%7B2%7D%7Bx%7D%5Cright%29%5Cleft%28%5Cfrac%7B%5Cleft%285%20x%20%2B%206%5Cright%29%5E%7B5%7D%20e%5E%7Bx%7D%20%5Ccos%5E%7B7%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B49%20x%5E%7B2%7D%20%5Cleft%285%20x%20-%202%5Cright%29%5E%7B8%7D%7D%20%5Cright%29%20%20%20" alt="LaTeX: y' = \left(- \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) " data-equation-content=" y' = \left(- \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) " />
Using some Trigonometric identities to simplify gives
<img class="equation_image" title=" y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{25}{5 x + 6}- \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) " src="/equation_images/%20y%27%20%3D%20%5Cleft%28-%207%20%5Ctan%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%201%20%2B%20%5Cfrac%7B25%7D%7B5%20x%20%2B%206%7D-%20%5Cfrac%7B40%7D%7B5%20x%20-%202%7D%20-%20%5Cfrac%7B2%7D%7Bx%7D%5Cright%29%5Cleft%28%5Cfrac%7B%5Cleft%285%20x%20%2B%206%5Cright%29%5E%7B5%7D%20e%5E%7Bx%7D%20%5Ccos%5E%7B7%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B49%20x%5E%7B2%7D%20%5Cleft%285%20x%20-%202%5Cright%29%5E%7B8%7D%7D%20%5Cright%29%20%20%20" alt="LaTeX: y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{25}{5 x + 6}- \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) " data-equation-content=" y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{25}{5 x + 6}- \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) " />
</p> </p>