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Find the derivative of y=(6x−7)4(7x+3)8ex(x−4)4(8x+4)5

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: ln⁡(y)=ln⁡((6x−7)4(7x+3)8ex(x−4)4(8x+4)5) Expanding the right hand side using the product and quotient properties of logarithms gives: ln⁡(y)=x+4ln⁡(6x−7)+8ln⁡(7x+3)−4ln⁡(x−4)−5ln⁡(8x+4)2 Taking the derivative on both sides of the equation yields: y′y=1−208x+4+567x+3+246x−7−4x−4 Solving for y′ and substituting out y using the original equation gives y′=(1−208x+4+567x+3+246x−7−4x−4)((6x−7)4(7x+3)8ex(x−4)4(8x+4)5)

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\begin{question}Find the derivative of $y = \frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}}$
    \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(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)}  \end{equation*}
Expanding the right hand side using the product and quotient properties of logarithms gives:
\begin{equation*}\ln(y) = x + 4 \ln{\left(6 x - 7 \right)} + 8 \ln{\left(7 x + 3 \right)}- 4 \ln{\left(x - 4 \right)} - \frac{5 \ln{\left(8 x + 4 \right)}}{2}  \end{equation*}
Taking the derivative on both sides of the equation yields:
\begin{equation*}\frac{y'}{y} = 1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}  \end{equation*}
Solving for $y'$ and substituting out y using the original equation gives
\begin{equation*}y' = \left(1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}\right)\left(\frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)  \end{equation*}
}

\end{question}

Download Question and Solution EnvironmentLATEX
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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = \frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B%5Cleft%286%20x%20-%207%5Cright%29%5E%7B4%7D%20%5Cleft%287%20x%20%2B%203%5Cright%29%5E%7B8%7D%20e%5E%7Bx%7D%7D%7B%5Cleft%28x%20-%204%5Cright%29%5E%7B4%7D%20%5Csqrt%7B%5Cleft%288%20x%20%2B%204%5Cright%29%5E%7B5%7D%7D%7D%20" alt="LaTeX:  \displaystyle y = \frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} " data-equation-content=" \displaystyle y = \frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} " /> </p> </p>
HTML for Canvas
<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(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)}   " src="/equation_images/%20%5Cln%28y%29%20%3D%20%5Cln%7B%5Cleft%28%5Cfrac%7B%5Cleft%286%20x%20-%207%5Cright%29%5E%7B4%7D%20%5Cleft%287%20x%20%2B%203%5Cright%29%5E%7B8%7D%20e%5E%7Bx%7D%7D%7B%5Cleft%28x%20-%204%5Cright%29%5E%7B4%7D%20%5Csqrt%7B%5Cleft%288%20x%20%2B%204%5Cright%29%5E%7B5%7D%7D%7D%20%5Cright%29%7D%20%20%20" alt="LaTeX:  \ln(y) = \ln{\left(\frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)}   " data-equation-content=" \ln(y) = \ln{\left(\frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)}   " /> 
Expanding the right hand side using the product and quotient properties of logarithms gives:
 <img class="equation_image" title=" \ln(y) = x + 4 \ln{\left(6 x - 7 \right)} + 8 \ln{\left(7 x + 3 \right)}- 4 \ln{\left(x - 4 \right)} - \frac{5 \ln{\left(8 x + 4 \right)}}{2}   " src="/equation_images/%20%5Cln%28y%29%20%3D%20x%20%2B%204%20%5Cln%7B%5Cleft%286%20x%20-%207%20%5Cright%29%7D%20%2B%208%20%5Cln%7B%5Cleft%287%20x%20%2B%203%20%5Cright%29%7D-%204%20%5Cln%7B%5Cleft%28x%20-%204%20%5Cright%29%7D%20-%20%5Cfrac%7B5%20%5Cln%7B%5Cleft%288%20x%20%2B%204%20%5Cright%29%7D%7D%7B2%7D%20%20%20" alt="LaTeX:  \ln(y) = x + 4 \ln{\left(6 x - 7 \right)} + 8 \ln{\left(7 x + 3 \right)}- 4 \ln{\left(x - 4 \right)} - \frac{5 \ln{\left(8 x + 4 \right)}}{2}   " data-equation-content=" \ln(y) = x + 4 \ln{\left(6 x - 7 \right)} + 8 \ln{\left(7 x + 3 \right)}- 4 \ln{\left(x - 4 \right)} - \frac{5 \ln{\left(8 x + 4 \right)}}{2}   " /> 
Taking the derivative on both sides of the equation yields:
 <img class="equation_image" title=" \frac{y'}{y} = 1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}   " src="/equation_images/%20%5Cfrac%7By%27%7D%7By%7D%20%3D%201%20-%20%5Cfrac%7B20%7D%7B8%20x%20%2B%204%7D%20%2B%20%5Cfrac%7B56%7D%7B7%20x%20%2B%203%7D%20%2B%20%5Cfrac%7B24%7D%7B6%20x%20-%207%7D%20-%20%5Cfrac%7B4%7D%7Bx%20-%204%7D%20%20%20" alt="LaTeX:  \frac{y'}{y} = 1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}   " data-equation-content=" \frac{y'}{y} = 1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}   " /> 
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(1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}\right)\left(\frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)   " src="/equation_images/%20y%27%20%3D%20%5Cleft%281%20-%20%5Cfrac%7B20%7D%7B8%20x%20%2B%204%7D%20%2B%20%5Cfrac%7B56%7D%7B7%20x%20%2B%203%7D%20%2B%20%5Cfrac%7B24%7D%7B6%20x%20-%207%7D%20-%20%5Cfrac%7B4%7D%7Bx%20-%204%7D%5Cright%29%5Cleft%28%5Cfrac%7B%5Cleft%286%20x%20-%207%5Cright%29%5E%7B4%7D%20%5Cleft%287%20x%20%2B%203%5Cright%29%5E%7B8%7D%20e%5E%7Bx%7D%7D%7B%5Cleft%28x%20-%204%5Cright%29%5E%7B4%7D%20%5Csqrt%7B%5Cleft%288%20x%20%2B%204%5Cright%29%5E%7B5%7D%7D%7D%20%5Cright%29%20%20%20" alt="LaTeX:  y' = \left(1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}\right)\left(\frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)   " data-equation-content=" y' = \left(1 - \frac{20}{8 x + 4} + \frac{56}{7 x + 3} + \frac{24}{6 x - 7} - \frac{4}{x - 4}\right)\left(\frac{\left(6 x - 7\right)^{4} \left(7 x + 3\right)^{8} e^{x}}{\left(x - 4\right)^{4} \sqrt{\left(8 x + 4\right)^{5}}} \right)   " /> 
</p> </p>