Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 2\right)^{7} \left(9 x - 9\right)^{6} e^{- x}}{\left(5 - 3 x\right)^{5} \left(5 x + 1\right)^{4}}

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 + 2\right)^{7} \left(9 x - 9\right)^{6} e^{- x}}{\left(5 - 3 x\right)^{5} \left(5 x + 1\right)^{4}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(x + 2 \right)} + 6 \ln{\left(9 x - 9 \right)}- x - 5 \ln{\left(5 - 3 x \right)} - 4 \ln{\left(5 x + 1 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{54}{9 x - 9} - \frac{20}{5 x + 1} + \frac{7}{x + 2} + \frac{15}{5 - 3 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{54}{9 x - 9} - \frac{20}{5 x + 1} + \frac{7}{x + 2} + \frac{15}{5 - 3 x}\right)\left(\frac{\left(x + 2\right)^{7} \left(9 x - 9\right)^{6} e^{- x}}{\left(5 - 3 x\right)^{5} \left(5 x + 1\right)^{4}} \right)