Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 4\right)^{3} \left(9 x - 5\right)^{6} e^{x}}{\left(x - 1\right)^{7} \sqrt{5 x + 4}}

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)^{3} \left(9 x - 5\right)^{6} e^{x}}{\left(x - 1\right)^{7} \sqrt{5 x + 4}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 3 \ln{\left(x + 4 \right)} + 6 \ln{\left(9 x - 5 \right)}- 7 \ln{\left(x - 1 \right)} - \frac{\ln{\left(5 x + 4 \right)}}{2} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{54}{9 x - 5} - \frac{5}{2 \left(5 x + 4\right)} + \frac{3}{x + 4} - \frac{7}{x - 1} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{54}{9 x - 5} - \frac{5}{2 \left(5 x + 4\right)} + \frac{3}{x + 4} - \frac{7}{x - 1}\right)\left(\frac{\left(x + 4\right)^{3} \left(9 x - 5\right)^{6} e^{x}}{\left(x - 1\right)^{7} \sqrt{5 x + 4}} \right)