Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 9\right)^{7} \left(8 x + 9\right)^{7} e^{- x} \sin^{2}{\left(x \right)}}{\left(- 6 x - 8\right)^{6} \left(7 x + 6\right)^{2} \sqrt{\left(x + 3\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(x - 9\right)^{7} \left(8 x + 9\right)^{7} e^{- x} \sin^{2}{\left(x \right)}}{\left(- 6 x - 8\right)^{6} \left(7 x + 6\right)^{2} \sqrt{\left(x + 3\right)^{7}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(x - 9 \right)} + 7 \ln{\left(8 x + 9 \right)} + 2 \ln{\left(\sin{\left(x \right)} \right)}- x - 6 \ln{\left(- 6 x - 8 \right)} - \frac{7 \ln{\left(x + 3 \right)}}{2} - 2 \ln{\left(7 x + 6 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{56}{8 x + 9} - \frac{14}{7 x + 6} - \frac{7}{2 \left(x + 3\right)} + \frac{7}{x - 9} + \frac{36}{- 6 x - 8} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{56}{8 x + 9} - \frac{14}{7 x + 6} - \frac{7}{2 \left(x + 3\right)} + \frac{7}{x - 9} + \frac{36}{- 6 x - 8}\right)\left(\frac{\left(x - 9\right)^{7} \left(8 x + 9\right)^{7} e^{- x} \sin^{2}{\left(x \right)}}{\left(- 6 x - 8\right)^{6} \left(7 x + 6\right)^{2} \sqrt{\left(x + 3\right)^{7}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{2}{\tan{\left(x \right)}} + \frac{56}{8 x + 9} + \frac{7}{x - 9}-1 - \frac{14}{7 x + 6} - \frac{7}{2 \left(x + 3\right)} + \frac{36}{- 6 x - 8}\right)\left(\frac{\left(x - 9\right)^{7} \left(8 x + 9\right)^{7} e^{- x} \sin^{2}{\left(x \right)}}{\left(- 6 x - 8\right)^{6} \left(7 x + 6\right)^{2} \sqrt{\left(x + 3\right)^{7}}} \right)