Find the derivative of LaTeX: \displaystyle y = \frac{x^{4} \left(- x - 6\right)^{7} e^{x}}{\left(x - 2\right)^{7} \left(3 x + 2\right)^{2} \sqrt{3 x + 7}}

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