Find the derivative of LaTeX: \displaystyle y = - \frac{\left(x - 6\right)^{3} \sqrt{9 x + 9} e^{- x}}{32768 x^{5} \left(- 8 x - 5\right)^{7}}

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 - 6\right)^{3} \sqrt{9 x + 9} e^{- x}}{32768 x^{5} \left(- 8 x - 5\right)^{7}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 3 \ln{\left(x - 6 \right)} + \frac{\ln{\left(9 x + 9 \right)}}{2}- x - 5 \ln{\left(x \right)} - 7 \ln{\left(- 8 x - 5 \right)} - 15 \ln{\left(2 \right)} - i \pi Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{9}{2 \left(9 x + 9\right)} + \frac{3}{x - 6} + \frac{56}{- 8 x - 5} - \frac{5}{x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{9}{2 \left(9 x + 9\right)} + \frac{3}{x - 6} + \frac{56}{- 8 x - 5} - \frac{5}{x}\right)\left(- \frac{\left(x - 6\right)^{3} \sqrt{9 x + 9} e^{- x}}{32768 x^{5} \left(- 8 x - 5\right)^{7}} \right)