Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 4\right)^{2} \sqrt{\left(5 x + 7\right)^{5}} e^{- x}}{\left(2 - 5 x\right)^{6} \left(7 x + 5\right)^{4} \left(9 x + 5\right)^{6}}

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 - 4\right)^{2} \sqrt{\left(5 x + 7\right)^{5}} e^{- x}}{\left(2 - 5 x\right)^{6} \left(7 x + 5\right)^{4} \left(9 x + 5\right)^{6}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 2 \ln{\left(x - 4 \right)} + \frac{5 \ln{\left(5 x + 7 \right)}}{2}- x - 6 \ln{\left(2 - 5 x \right)} - 4 \ln{\left(7 x + 5 \right)} - 6 \ln{\left(9 x + 5 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 - \frac{54}{9 x + 5} - \frac{28}{7 x + 5} + \frac{25}{2 \left(5 x + 7\right)} + \frac{2}{x - 4} + \frac{30}{2 - 5 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 - \frac{54}{9 x + 5} - \frac{28}{7 x + 5} + \frac{25}{2 \left(5 x + 7\right)} + \frac{2}{x - 4} + \frac{30}{2 - 5 x}\right)\left(\frac{\left(x - 4\right)^{2} \sqrt{\left(5 x + 7\right)^{5}} e^{- x}}{\left(2 - 5 x\right)^{6} \left(7 x + 5\right)^{4} \left(9 x + 5\right)^{6}} \right)