Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 6\right)^{2} \sqrt{\left(7 x + 4\right)^{3}} e^{- x}}{\left(2 x - 3\right)^{4} \left(8 x + 6\right)^{2}}

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 - 6\right)^{2} \sqrt{\left(7 x + 4\right)^{3}} e^{- x}}{\left(2 x - 3\right)^{4} \left(8 x + 6\right)^{2}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 2 \ln{\left(x - 6 \right)} + \frac{3 \ln{\left(7 x + 4 \right)}}{2}- x - 4 \ln{\left(2 x - 3 \right)} - 2 \ln{\left(8 x + 6 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 - \frac{16}{8 x + 6} + \frac{21}{2 \left(7 x + 4\right)} - \frac{8}{2 x - 3} + \frac{2}{x - 6} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 - \frac{16}{8 x + 6} + \frac{21}{2 \left(7 x + 4\right)} - \frac{8}{2 x - 3} + \frac{2}{x - 6}\right)\left(\frac{\left(x - 6\right)^{2} \sqrt{\left(7 x + 4\right)^{3}} e^{- x}}{\left(2 x - 3\right)^{4} \left(8 x + 6\right)^{2}} \right)