Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 6 x - 4\right)^{3} \left(x + 7\right)^{6} e^{x}}{\left(- 7 x - 4\right)^{6} \left(2 x + 8\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(- 6 x - 4\right)^{3} \left(x + 7\right)^{6} e^{x}}{\left(- 7 x - 4\right)^{6} \left(2 x + 8\right)^{7}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 3 \ln{\left(- 6 x - 4 \right)} + 6 \ln{\left(x + 7 \right)}- 6 \ln{\left(- 7 x - 4 \right)} - 7 \ln{\left(2 x + 8 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 - \frac{14}{2 x + 8} + \frac{6}{x + 7} - \frac{18}{- 6 x - 4} + \frac{42}{- 7 x - 4} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 - \frac{14}{2 x + 8} + \frac{6}{x + 7} - \frac{18}{- 6 x - 4} + \frac{42}{- 7 x - 4}\right)\left(\frac{\left(- 6 x - 4\right)^{3} \left(x + 7\right)^{6} e^{x}}{\left(- 7 x - 4\right)^{6} \left(2 x + 8\right)^{7}} \right)