Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 9 x - 3\right)^{8} \left(x + 9\right)^{8}}{\left(6 x + 3\right)^{7} \sqrt{\left(3 x + 7\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(- 9 x - 3\right)^{8} \left(x + 9\right)^{8}}{\left(6 x + 3\right)^{7} \sqrt{\left(3 x + 7\right)^{7}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 8 \ln{\left(- 9 x - 3 \right)} + 8 \ln{\left(x + 9 \right)}- \frac{7 \ln{\left(3 x + 7 \right)}}{2} - 7 \ln{\left(6 x + 3 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{42}{6 x + 3} - \frac{21}{2 \left(3 x + 7\right)} + \frac{8}{x + 9} - \frac{72}{- 9 x - 3} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{42}{6 x + 3} - \frac{21}{2 \left(3 x + 7\right)} + \frac{8}{x + 9} - \frac{72}{- 9 x - 3}\right)\left(\frac{\left(- 9 x - 3\right)^{8} \left(x + 9\right)^{8}}{\left(6 x + 3\right)^{7} \sqrt{\left(3 x + 7\right)^{7}}} \right)