Find the derivative of $LaTeX: \displaystyle y = \frac{\left(- 5 x - 9\right)^{6} \left(x + 1\right)^{6} e^{x}}{\left(4 x + 1\right)^{4} \sqrt{\left(4 x + 8\right)^{3}}}$

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: $LaTeX: \ln(y) = \ln{\left(\frac{\left(- 5 x - 9\right)^{6} \left(x + 1\right)^{6} e^{x}}{\left(4 x + 1\right)^{4} \sqrt{\left(4 x + 8\right)^{3}}} \right)}$ Expanding the right hand side using the product and quotient properties of logarithms gives: $LaTeX: \ln(y) = x + 6 \ln{\left(- 5 x - 9 \right)} + 6 \ln{\left(x + 1 \right)}- 4 \ln{\left(4 x + 1 \right)} - \frac{3 \ln{\left(4 x + 8 \right)}}{2}$ Taking the derivative on both sides of the equation yields: $LaTeX: \frac{y'}{y} = 1 - \frac{6}{4 x + 8} - \frac{16}{4 x + 1} + \frac{6}{x + 1} - \frac{30}{- 5 x - 9}$ Solving for $LaTeX: \displaystyle y'$ and substituting out y using the original equation gives $LaTeX: y' = \left(1 - \frac{6}{4 x + 8} - \frac{16}{4 x + 1} + \frac{6}{x + 1} - \frac{30}{- 5 x - 9}\right)\left(\frac{\left(- 5 x - 9\right)^{6} \left(x + 1\right)^{6} e^{x}}{\left(4 x + 1\right)^{4} \sqrt{\left(4 x + 8\right)^{3}}} \right)$