Find the derivative of LaTeX: \displaystyle y = \frac{\left(7 x + 7\right)^{4} \sqrt{\left(6 x + 7\right)^{7}} e^{- x}}{x^{2} \left(2 x - 4\right)^{5} \left(7 x - 5\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(7 x + 7\right)^{4} \sqrt{\left(6 x + 7\right)^{7}} e^{- x}}{x^{2} \left(2 x - 4\right)^{5} \left(7 x - 5\right)^{5}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{7 \ln{\left(6 x + 7 \right)}}{2} + 4 \ln{\left(7 x + 7 \right)}- x - 2 \ln{\left(x \right)} - 5 \ln{\left(2 x - 4 \right)} - 5 \ln{\left(7 x - 5 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{28}{7 x + 7} - \frac{35}{7 x - 5} + \frac{21}{6 x + 7} - \frac{10}{2 x - 4} - \frac{2}{x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{28}{7 x + 7} - \frac{35}{7 x - 5} + \frac{21}{6 x + 7} - \frac{10}{2 x - 4} - \frac{2}{x}\right)\left(\frac{\left(7 x + 7\right)^{4} \sqrt{\left(6 x + 7\right)^{7}} e^{- x}}{x^{2} \left(2 x - 4\right)^{5} \left(7 x - 5\right)^{5}} \right)