Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 8 x - 8\right)^{5} e^{x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(- 4 x - 3\right)^{8} \left(9 x - 5\right)^{7} \sqrt{\left(5 x + 8\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(- 8 x - 8\right)^{5} e^{x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(- 4 x - 3\right)^{8} \left(9 x - 5\right)^{7} \sqrt{\left(5 x + 8\right)^{7}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 5 \ln{\left(- 8 x - 8 \right)} + 2 \ln{\left(\sin{\left(x \right)} \right)} + 6 \ln{\left(\cos{\left(x \right)} \right)}- 8 \ln{\left(- 4 x - 3 \right)} - \frac{7 \ln{\left(5 x + 8 \right)}}{2} - 7 \ln{\left(9 x - 5 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{63}{9 x - 5} - \frac{35}{2 \left(5 x + 8\right)} + \frac{32}{- 4 x - 3} - \frac{40}{- 8 x - 8} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{63}{9 x - 5} - \frac{35}{2 \left(5 x + 8\right)} + \frac{32}{- 4 x - 3} - \frac{40}{- 8 x - 8}\right)\left(\frac{\left(- 8 x - 8\right)^{5} e^{x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(- 4 x - 3\right)^{8} \left(9 x - 5\right)^{7} \sqrt{\left(5 x + 8\right)^{7}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 6 \tan{\left(x \right)} + 1 + \frac{2}{\tan{\left(x \right)}} - \frac{40}{- 8 x - 8}- \frac{63}{9 x - 5} - \frac{35}{2 \left(5 x + 8\right)} + \frac{32}{- 4 x - 3}\right)\left(\frac{\left(- 8 x - 8\right)^{5} e^{x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(- 4 x - 3\right)^{8} \left(9 x - 5\right)^{7} \sqrt{\left(5 x + 8\right)^{7}}} \right)