Find the derivative of LaTeX: \displaystyle y = \frac{\left(2 - 8 x\right)^{2} \left(x - 2\right)^{5} \left(9 x - 5\right)^{2} e^{x}}{\left(5 - 2 x\right)^{5} \sqrt{3 x + 9}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(2 - 8 x\right)^{2} \left(x - 2\right)^{5} \left(9 x - 5\right)^{2} e^{x}}{\left(5 - 2 x\right)^{5} \sqrt{3 x + 9}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 2 \ln{\left(2 - 8 x \right)} + 5 \ln{\left(x - 2 \right)} + 2 \ln{\left(9 x - 5 \right)}- 5 \ln{\left(5 - 2 x \right)} - \frac{\ln{\left(3 x + 9 \right)}}{2} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{18}{9 x - 5} - \frac{3}{2 \left(3 x + 9\right)} + \frac{5}{x - 2} + \frac{10}{5 - 2 x} - \frac{16}{2 - 8 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{18}{9 x - 5} - \frac{3}{2 \left(3 x + 9\right)} + \frac{5}{x - 2} + \frac{10}{5 - 2 x} - \frac{16}{2 - 8 x}\right)\left(\frac{\left(2 - 8 x\right)^{2} \left(x - 2\right)^{5} \left(9 x - 5\right)^{2} e^{x}}{\left(5 - 2 x\right)^{5} \sqrt{3 x + 9}} \right)