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Find the derivative of \(\displaystyle y = \frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}}\)
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(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right)} \end{equation*} Expanding the right hand side using the product and quotient properties of logarithms gives: \begin{equation*}\ln(y) = 6 \ln{\left(x - 6 \right)} + 5 \ln{\left(7 x - 1 \right)} + 8 \ln{\left(\cos{\left(x \right)} \right)}- \frac{5 \ln{\left(4 x + 6 \right)}}{2} - 8 \ln{\left(\sin{\left(x \right)} \right)} \end{equation*} Taking the derivative on both sides of the equation yields: \begin{equation*}\frac{y'}{y} = - \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6} \end{equation*} Solving for \(\displaystyle y'\) and substituting out y using the original equation gives \begin{equation*}y' = \left(- \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) \end{equation*} Using some Trigonometric identities to simplify gives \begin{equation*}y' = \left(- 8 \tan{\left(x \right)} + \frac{35}{7 x - 1} + \frac{6}{x - 6}- \frac{8}{\tan{\left(x \right)}} - \frac{10}{4 x + 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) \end{equation*}
\begin{question}Find the derivative of $y = \frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}}$
\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(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right)} \end{equation*}
Expanding the right hand side using the product and quotient properties of logarithms gives:
\begin{equation*}\ln(y) = 6 \ln{\left(x - 6 \right)} + 5 \ln{\left(7 x - 1 \right)} + 8 \ln{\left(\cos{\left(x \right)} \right)}- \frac{5 \ln{\left(4 x + 6 \right)}}{2} - 8 \ln{\left(\sin{\left(x \right)} \right)} \end{equation*}
Taking the derivative on both sides of the equation yields:
\begin{equation*}\frac{y'}{y} = - \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6} \end{equation*}
Solving for $y'$ and substituting out y using the original equation gives
\begin{equation*}y' = \left(- \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) \end{equation*}
Using some Trigonometric identities to simplify gives
\begin{equation*}y' = \left(- 8 \tan{\left(x \right)} + \frac{35}{7 x - 1} + \frac{6}{x - 6}- \frac{8}{\tan{\left(x \right)}} - \frac{10}{4 x + 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) \end{equation*}
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\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(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B%5Cleft%28x%20-%206%5Cright%29%5E%7B6%7D%20%5Cleft%287%20x%20-%201%5Cright%29%5E%7B5%7D%20%5Ccos%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csqrt%7B%5Cleft%284%20x%20%2B%206%5Cright%29%5E%7B5%7D%7D%20%5Csin%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX: \displaystyle y = \frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} " data-equation-content=" \displaystyle y = \frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} " /> </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(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right)} " src="/equation_images/%20%5Cln%28y%29%20%3D%20%5Cln%7B%5Cleft%28%5Cfrac%7B%5Cleft%28x%20-%206%5Cright%29%5E%7B6%7D%20%5Cleft%287%20x%20-%201%5Cright%29%5E%7B5%7D%20%5Ccos%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csqrt%7B%5Cleft%284%20x%20%2B%206%5Cright%29%5E%7B5%7D%7D%20%5Csin%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%5Cright%29%7D%20%20%20" alt="LaTeX: \ln(y) = \ln{\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right)} " data-equation-content=" \ln(y) = \ln{\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right)} " />
Expanding the right hand side using the product and quotient properties of logarithms gives:
<img class="equation_image" title=" \ln(y) = 6 \ln{\left(x - 6 \right)} + 5 \ln{\left(7 x - 1 \right)} + 8 \ln{\left(\cos{\left(x \right)} \right)}- \frac{5 \ln{\left(4 x + 6 \right)}}{2} - 8 \ln{\left(\sin{\left(x \right)} \right)} " src="/equation_images/%20%5Cln%28y%29%20%3D%206%20%5Cln%7B%5Cleft%28x%20-%206%20%5Cright%29%7D%20%2B%205%20%5Cln%7B%5Cleft%287%20x%20-%201%20%5Cright%29%7D%20%2B%208%20%5Cln%7B%5Cleft%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cright%29%7D-%20%5Cfrac%7B5%20%5Cln%7B%5Cleft%284%20x%20%2B%206%20%5Cright%29%7D%7D%7B2%7D%20-%208%20%5Cln%7B%5Cleft%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cright%29%7D%20%20%20" alt="LaTeX: \ln(y) = 6 \ln{\left(x - 6 \right)} + 5 \ln{\left(7 x - 1 \right)} + 8 \ln{\left(\cos{\left(x \right)} \right)}- \frac{5 \ln{\left(4 x + 6 \right)}}{2} - 8 \ln{\left(\sin{\left(x \right)} \right)} " data-equation-content=" \ln(y) = 6 \ln{\left(x - 6 \right)} + 5 \ln{\left(7 x - 1 \right)} + 8 \ln{\left(\cos{\left(x \right)} \right)}- \frac{5 \ln{\left(4 x + 6 \right)}}{2} - 8 \ln{\left(\sin{\left(x \right)} \right)} " />
Taking the derivative on both sides of the equation yields:
<img class="equation_image" title=" \frac{y'}{y} = - \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6} " src="/equation_images/%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20-%20%5Cfrac%7B8%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%20-%20%5Cfrac%7B8%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%2B%20%5Cfrac%7B35%7D%7B7%20x%20-%201%7D%20-%20%5Cfrac%7B10%7D%7B4%20x%20%2B%206%7D%20%2B%20%5Cfrac%7B6%7D%7Bx%20-%206%7D%20%20%20" alt="LaTeX: \frac{y'}{y} = - \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6} " data-equation-content=" \frac{y'}{y} = - \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6} " />
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{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) " src="/equation_images/%20y%27%20%3D%20%5Cleft%28-%20%5Cfrac%7B8%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%20-%20%5Cfrac%7B8%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%2B%20%5Cfrac%7B35%7D%7B7%20x%20-%201%7D%20-%20%5Cfrac%7B10%7D%7B4%20x%20%2B%206%7D%20%2B%20%5Cfrac%7B6%7D%7Bx%20-%206%7D%5Cright%29%5Cleft%28%5Cfrac%7B%5Cleft%28x%20-%206%5Cright%29%5E%7B6%7D%20%5Cleft%287%20x%20-%201%5Cright%29%5E%7B5%7D%20%5Ccos%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csqrt%7B%5Cleft%284%20x%20%2B%206%5Cright%29%5E%7B5%7D%7D%20%5Csin%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%5Cright%29%20%20%20" alt="LaTeX: y' = \left(- \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) " data-equation-content=" y' = \left(- \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{35}{7 x - 1} - \frac{10}{4 x + 6} + \frac{6}{x - 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) " />
Using some Trigonometric identities to simplify gives
<img class="equation_image" title=" y' = \left(- 8 \tan{\left(x \right)} + \frac{35}{7 x - 1} + \frac{6}{x - 6}- \frac{8}{\tan{\left(x \right)}} - \frac{10}{4 x + 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) " src="/equation_images/%20y%27%20%3D%20%5Cleft%28-%208%20%5Ctan%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B35%7D%7B7%20x%20-%201%7D%20%2B%20%5Cfrac%7B6%7D%7Bx%20-%206%7D-%20%5Cfrac%7B8%7D%7B%5Ctan%7B%5Cleft%28x%20%5Cright%29%7D%7D%20-%20%5Cfrac%7B10%7D%7B4%20x%20%2B%206%7D%5Cright%29%5Cleft%28%5Cfrac%7B%5Cleft%28x%20-%206%5Cright%29%5E%7B6%7D%20%5Cleft%287%20x%20-%201%5Cright%29%5E%7B5%7D%20%5Ccos%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B%5Csqrt%7B%5Cleft%284%20x%20%2B%206%5Cright%29%5E%7B5%7D%7D%20%5Csin%5E%7B8%7D%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%5Cright%29%20%20%20" alt="LaTeX: y' = \left(- 8 \tan{\left(x \right)} + \frac{35}{7 x - 1} + \frac{6}{x - 6}- \frac{8}{\tan{\left(x \right)}} - \frac{10}{4 x + 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) " data-equation-content=" y' = \left(- 8 \tan{\left(x \right)} + \frac{35}{7 x - 1} + \frac{6}{x - 6}- \frac{8}{\tan{\left(x \right)}} - \frac{10}{4 x + 6}\right)\left(\frac{\left(x - 6\right)^{6} \left(7 x - 1\right)^{5} \cos^{8}{\left(x \right)}}{\sqrt{\left(4 x + 6\right)^{5}} \sin^{8}{\left(x \right)}} \right) " />
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