Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 9 x - 8\right)^{8} \left(x - 8\right)^{6} \left(8 x + 1\right)^{4} e^{- x}}{\left(- 4 x - 2\right)^{4} \sqrt{\left(9 x + 6\right)^{7}} \sin^{7}{\left(x \right)}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(- 9 x - 8\right)^{8} \left(x - 8\right)^{6} \left(8 x + 1\right)^{4} e^{- x}}{\left(- 4 x - 2\right)^{4} \sqrt{\left(9 x + 6\right)^{7}} \sin^{7}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 8 \ln{\left(- 9 x - 8 \right)} + 6 \ln{\left(x - 8 \right)} + 4 \ln{\left(8 x + 1 \right)}- x - 4 \ln{\left(- 4 x - 2 \right)} - \frac{7 \ln{\left(9 x + 6 \right)}}{2} - 7 \ln{\left(\sin{\left(x \right)} \right)} 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{63}{2 \left(9 x + 6\right)} + \frac{32}{8 x + 1} + \frac{6}{x - 8} + \frac{16}{- 4 x - 2} - \frac{72}{- 9 x - 8} 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{63}{2 \left(9 x + 6\right)} + \frac{32}{8 x + 1} + \frac{6}{x - 8} + \frac{16}{- 4 x - 2} - \frac{72}{- 9 x - 8}\right)\left(\frac{\left(- 9 x - 8\right)^{8} \left(x - 8\right)^{6} \left(8 x + 1\right)^{4} e^{- x}}{\left(- 4 x - 2\right)^{4} \sqrt{\left(9 x + 6\right)^{7}} \sin^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{32}{8 x + 1} + \frac{6}{x - 8} - \frac{72}{- 9 x - 8}-1 - \frac{7}{\tan{\left(x \right)}} - \frac{63}{2 \left(9 x + 6\right)} + \frac{16}{- 4 x - 2}\right)\left(\frac{\left(- 9 x - 8\right)^{8} \left(x - 8\right)^{6} \left(8 x + 1\right)^{4} e^{- x}}{\left(- 4 x - 2\right)^{4} \sqrt{\left(9 x + 6\right)^{7}} \sin^{7}{\left(x \right)}} \right)