Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 9\right)^{4} \sqrt{\left(3 x + 3\right)^{5}} e^{- x} \cos^{8}{\left(x \right)}}{\left(3 - 5 x\right)^{4} \left(9 - 8 x\right)^{5} \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(x + 9\right)^{4} \sqrt{\left(3 x + 3\right)^{5}} e^{- x} \cos^{8}{\left(x \right)}}{\left(3 - 5 x\right)^{4} \left(9 - 8 x\right)^{5} \sin^{7}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 4 \ln{\left(x + 9 \right)} + \frac{5 \ln{\left(3 x + 3 \right)}}{2} + 8 \ln{\left(\cos{\left(x \right)} \right)}- x - 4 \ln{\left(3 - 5 x \right)} - 5 \ln{\left(9 - 8 x \right)} - 7 \ln{\left(\sin{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 - \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{15}{2 \left(3 x + 3\right)} + \frac{4}{x + 9} + \frac{40}{9 - 8 x} + \frac{20}{3 - 5 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 - \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{15}{2 \left(3 x + 3\right)} + \frac{4}{x + 9} + \frac{40}{9 - 8 x} + \frac{20}{3 - 5 x}\right)\left(\frac{\left(x + 9\right)^{4} \sqrt{\left(3 x + 3\right)^{5}} e^{- x} \cos^{8}{\left(x \right)}}{\left(3 - 5 x\right)^{4} \left(9 - 8 x\right)^{5} \sin^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 8 \tan{\left(x \right)} + \frac{15}{2 \left(3 x + 3\right)} + \frac{4}{x + 9}-1 - \frac{7}{\tan{\left(x \right)}} + \frac{40}{9 - 8 x} + \frac{20}{3 - 5 x}\right)\left(\frac{\left(x + 9\right)^{4} \sqrt{\left(3 x + 3\right)^{5}} e^{- x} \cos^{8}{\left(x \right)}}{\left(3 - 5 x\right)^{4} \left(9 - 8 x\right)^{5} \sin^{7}{\left(x \right)}} \right)