Find the derivative of $LaTeX: \displaystyle y = \frac{\left(- 3 x - 8\right)^{7} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 9 x - 4\right)^{6} \sqrt{\left(6 x + 4\right)^{3}}}$

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: $LaTeX: \ln(y) = \ln{\left(\frac{\left(- 3 x - 8\right)^{7} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 9 x - 4\right)^{6} \sqrt{\left(6 x + 4\right)^{3}}} \right)}$ Expanding the right hand side using the product and quotient properties of logarithms gives: $LaTeX: \ln(y) = 7 \ln{\left(- 3 x - 8 \right)} + 7 \ln{\left(\sin{\left(x \right)} \right)}- x - 6 \ln{\left(- 9 x - 4 \right)} - \frac{3 \ln{\left(6 x + 4 \right)}}{2}$ Taking the derivative on both sides of the equation yields: $LaTeX: \frac{y'}{y} = -1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{9}{6 x + 4} - \frac{21}{- 3 x - 8} + \frac{54}{- 9 x - 4}$ Solving for $LaTeX: \displaystyle y'$ and substituting out y using the original equation gives $LaTeX: y' = \left(-1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{9}{6 x + 4} - \frac{21}{- 3 x - 8} + \frac{54}{- 9 x - 4}\right)\left(\frac{\left(- 3 x - 8\right)^{7} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 9 x - 4\right)^{6} \sqrt{\left(6 x + 4\right)^{3}}} \right)$ Using some Trigonometric identities to simplify gives $LaTeX: y' = \left(\frac{7}{\tan{\left(x \right)}} - \frac{21}{- 3 x - 8}-1 - \frac{9}{6 x + 4} + \frac{54}{- 9 x - 4}\right)\left(\frac{\left(- 3 x - 8\right)^{7} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 9 x - 4\right)^{6} \sqrt{\left(6 x + 4\right)^{3}}} \right)$