Find the derivative of LaTeX: \displaystyle y = \frac{\left(7 - 2 x\right)^{2} \sqrt{\left(5 x + 4\right)^{5}} e^{x}}{\left(6 - 7 x\right)^{7} \left(x + 4\right)^{8}}

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 - 2 x\right)^{2} \sqrt{\left(5 x + 4\right)^{5}} e^{x}}{\left(6 - 7 x\right)^{7} \left(x + 4\right)^{8}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 2 \ln{\left(7 - 2 x \right)} + \frac{5 \ln{\left(5 x + 4 \right)}}{2}- 7 \ln{\left(6 - 7 x \right)} - 8 \ln{\left(x + 4 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{25}{2 \left(5 x + 4\right)} - \frac{8}{x + 4} - \frac{4}{7 - 2 x} + \frac{49}{6 - 7 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{25}{2 \left(5 x + 4\right)} - \frac{8}{x + 4} - \frac{4}{7 - 2 x} + \frac{49}{6 - 7 x}\right)\left(\frac{\left(7 - 2 x\right)^{2} \sqrt{\left(5 x + 4\right)^{5}} e^{x}}{\left(6 - 7 x\right)^{7} \left(x + 4\right)^{8}} \right)