Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 9\right)^{7} \sqrt{\left(5 x + 5\right)^{5}}}{\left(4 x + 6\right)^{8} \left(7 x - 1\right)^{5}}

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} \sqrt{\left(5 x + 5\right)^{5}}}{\left(4 x + 6\right)^{8} \left(7 x - 1\right)^{5}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(x + 9 \right)} + \frac{5 \ln{\left(5 x + 5 \right)}}{2}- 8 \ln{\left(4 x + 6 \right)} - 5 \ln{\left(7 x - 1 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{35}{7 x - 1} + \frac{25}{2 \left(5 x + 5\right)} - \frac{32}{4 x + 6} + \frac{7}{x + 9} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{35}{7 x - 1} + \frac{25}{2 \left(5 x + 5\right)} - \frac{32}{4 x + 6} + \frac{7}{x + 9}\right)\left(\frac{\left(x + 9\right)^{7} \sqrt{\left(5 x + 5\right)^{5}}}{\left(4 x + 6\right)^{8} \left(7 x - 1\right)^{5}} \right)