Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 7 x - 6\right)^{8} \left(5 x - 8\right)^{3} \left(9 x + 2\right)^{7} e^{x}}{\sqrt{\left(3 x + 1\right)^{7}} \sin^{5}{\left(x \right)} \cos^{5}{\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(- 7 x - 6\right)^{8} \left(5 x - 8\right)^{3} \left(9 x + 2\right)^{7} e^{x}}{\sqrt{\left(3 x + 1\right)^{7}} \sin^{5}{\left(x \right)} \cos^{5}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 8 \ln{\left(- 7 x - 6 \right)} + 3 \ln{\left(5 x - 8 \right)} + 7 \ln{\left(9 x + 2 \right)}- \frac{7 \ln{\left(3 x + 1 \right)}}{2} - 5 \ln{\left(\sin{\left(x \right)} \right)} - 5 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{63}{9 x + 2} + \frac{15}{5 x - 8} - \frac{21}{2 \left(3 x + 1\right)} - \frac{56}{- 7 x - 6} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{63}{9 x + 2} + \frac{15}{5 x - 8} - \frac{21}{2 \left(3 x + 1\right)} - \frac{56}{- 7 x - 6}\right)\left(\frac{\left(- 7 x - 6\right)^{8} \left(5 x - 8\right)^{3} \left(9 x + 2\right)^{7} e^{x}}{\sqrt{\left(3 x + 1\right)^{7}} \sin^{5}{\left(x \right)} \cos^{5}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{63}{9 x + 2} + \frac{15}{5 x - 8} - \frac{56}{- 7 x - 6}5 \tan{\left(x \right)} - \frac{5}{\tan{\left(x \right)}} - \frac{21}{2 \left(3 x + 1\right)}\right)\left(\frac{\left(- 7 x - 6\right)^{8} \left(5 x - 8\right)^{3} \left(9 x + 2\right)^{7} e^{x}}{\sqrt{\left(3 x + 1\right)^{7}} \sin^{5}{\left(x \right)} \cos^{5}{\left(x \right)}} \right)