Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 2\right)^{7} \left(5 x + 8\right)^{7} \left(8 x - 6\right)^{7} e^{x}}{\left(7 x - 3\right)^{7} \sqrt{\left(x + 9\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(x + 2\right)^{7} \left(5 x + 8\right)^{7} \left(8 x - 6\right)^{7} e^{x}}{\left(7 x - 3\right)^{7} \sqrt{\left(x + 9\right)^{3}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 7 \ln{\left(x + 2 \right)} + 7 \ln{\left(5 x + 8 \right)} + 7 \ln{\left(8 x - 6 \right)}- \frac{3 \ln{\left(x + 9 \right)}}{2} - 7 \ln{\left(7 x - 3 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{56}{8 x - 6} - \frac{49}{7 x - 3} + \frac{35}{5 x + 8} - \frac{3}{2 \left(x + 9\right)} + \frac{7}{x + 2} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{56}{8 x - 6} - \frac{49}{7 x - 3} + \frac{35}{5 x + 8} - \frac{3}{2 \left(x + 9\right)} + \frac{7}{x + 2}\right)\left(\frac{\left(x + 2\right)^{7} \left(5 x + 8\right)^{7} \left(8 x - 6\right)^{7} e^{x}}{\left(7 x - 3\right)^{7} \sqrt{\left(x + 9\right)^{3}}} \right)