Find the derivative of $LaTeX: \displaystyle y = \frac{\left(4 x - 3\right)^{6} \left(9 x + 3\right)^{2}}{x^{6} \sqrt{\left(x + 7\right)^{7}}}$

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: $LaTeX: \ln(y) = \ln{\left(\frac{\left(4 x - 3\right)^{6} \left(9 x + 3\right)^{2}}{x^{6} \sqrt{\left(x + 7\right)^{7}}} \right)}$ Expanding the right hand side using the product and quotient properties of logarithms gives: $LaTeX: \ln(y) = 6 \ln{\left(4 x - 3 \right)} + 2 \ln{\left(9 x + 3 \right)}- 6 \ln{\left(x \right)} - \frac{7 \ln{\left(x + 7 \right)}}{2}$ Taking the derivative on both sides of the equation yields: $LaTeX: \frac{y'}{y} = \frac{18}{9 x + 3} + \frac{24}{4 x - 3} - \frac{7}{2 \left(x + 7\right)} - \frac{6}{x}$ Solving for $LaTeX: \displaystyle y'$ and substituting out y using the original equation gives $LaTeX: y' = \left(\frac{18}{9 x + 3} + \frac{24}{4 x - 3} - \frac{7}{2 \left(x + 7\right)} - \frac{6}{x}\right)\left(\frac{\left(4 x - 3\right)^{6} \left(9 x + 3\right)^{2}}{x^{6} \sqrt{\left(x + 7\right)^{7}}} \right)$