Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 9\right)^{7} \left(7 x + 2\right)^{2} e^{- x}}{\left(- 8 x - 5\right)^{2} \left(6 x - 2\right)^{6} \sqrt{\left(3 x + 5\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(x - 9\right)^{7} \left(7 x + 2\right)^{2} e^{- x}}{\left(- 8 x - 5\right)^{2} \left(6 x - 2\right)^{6} \sqrt{\left(3 x + 5\right)^{5}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(x - 9 \right)} + 2 \ln{\left(7 x + 2 \right)}- x - 2 \ln{\left(- 8 x - 5 \right)} - \frac{5 \ln{\left(3 x + 5 \right)}}{2} - 6 \ln{\left(6 x - 2 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{14}{7 x + 2} - \frac{36}{6 x - 2} - \frac{15}{2 \left(3 x + 5\right)} + \frac{7}{x - 9} + \frac{16}{- 8 x - 5} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{14}{7 x + 2} - \frac{36}{6 x - 2} - \frac{15}{2 \left(3 x + 5\right)} + \frac{7}{x - 9} + \frac{16}{- 8 x - 5}\right)\left(\frac{\left(x - 9\right)^{7} \left(7 x + 2\right)^{2} e^{- x}}{\left(- 8 x - 5\right)^{2} \left(6 x - 2\right)^{6} \sqrt{\left(3 x + 5\right)^{5}}} \right)