Find the derivative of 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}}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: 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)} Expanding the right hand side using the product and quotient properties of logarithms gives: 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} Taking the derivative on both sides of the equation yields: LaTeX: \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 LaTeX: \displaystyle y' and substituting out y using the original equation gives 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)