Find the derivative of LaTeX: \displaystyle y = \frac{\left(4 - 5 x\right)^{7} \sqrt{\left(3 x + 6\right)^{7}} e^{- x}}{\left(x + 7\right)^{7} \left(4 x - 1\right)^{3}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(4 - 5 x\right)^{7} \sqrt{\left(3 x + 6\right)^{7}} e^{- x}}{\left(x + 7\right)^{7} \left(4 x - 1\right)^{3}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(4 - 5 x \right)} + \frac{7 \ln{\left(3 x + 6 \right)}}{2}- x - 7 \ln{\left(x + 7 \right)} - 3 \ln{\left(4 x - 1 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 - \frac{12}{4 x - 1} + \frac{21}{2 \left(3 x + 6\right)} - \frac{7}{x + 7} - \frac{35}{4 - 5 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 - \frac{12}{4 x - 1} + \frac{21}{2 \left(3 x + 6\right)} - \frac{7}{x + 7} - \frac{35}{4 - 5 x}\right)\left(\frac{\left(4 - 5 x\right)^{7} \sqrt{\left(3 x + 6\right)^{7}} e^{- x}}{\left(x + 7\right)^{7} \left(4 x - 1\right)^{3}} \right)