Find the derivative of LaTeX: \displaystyle y = \frac{\left(7 x - 7\right)^{4} \left(7 x - 6\right)^{2} e^{x}}{\left(x + 5\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(7 x - 7\right)^{4} \left(7 x - 6\right)^{2} e^{x}}{\left(x + 5\right)^{3}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 4 \ln{\left(7 x - 7 \right)} + 2 \ln{\left(7 x - 6 \right)}- 3 \ln{\left(x + 5 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{14}{7 x - 6} + \frac{28}{7 x - 7} - \frac{3}{x + 5} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{14}{7 x - 6} + \frac{28}{7 x - 7} - \frac{3}{x + 5}\right)\left(\frac{\left(7 x - 7\right)^{4} \left(7 x - 6\right)^{2} e^{x}}{\left(x + 5\right)^{3}} \right)