Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 9\right)^{7} \left(9 x + 7\right)^{3} e^{x}}{\sqrt{3 x + 2} \left(9 x - 1\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 - 9\right)^{7} \left(9 x + 7\right)^{3} e^{x}}{\sqrt{3 x + 2} \left(9 x - 1\right)^{7}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 7 \ln{\left(x - 9 \right)} + 3 \ln{\left(9 x + 7 \right)}- \frac{\ln{\left(3 x + 2 \right)}}{2} - 7 \ln{\left(9 x - 1 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{27}{9 x + 7} - \frac{63}{9 x - 1} - \frac{3}{2 \left(3 x + 2\right)} + \frac{7}{x - 9} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{27}{9 x + 7} - \frac{63}{9 x - 1} - \frac{3}{2 \left(3 x + 2\right)} + \frac{7}{x - 9}\right)\left(\frac{\left(x - 9\right)^{7} \left(9 x + 7\right)^{3} e^{x}}{\sqrt{3 x + 2} \left(9 x - 1\right)^{7}} \right)