Find the derivative of LaTeX: \displaystyle y = \frac{\left(2 - 4 x\right)^{5} \left(3 x + 5\right)^{7} e^{- x} \cos^{8}{\left(x \right)}}{\left(x + 7\right)^{6} \left(3 x - 4\right)^{2} \sin^{3}{\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(2 - 4 x\right)^{5} \left(3 x + 5\right)^{7} e^{- x} \cos^{8}{\left(x \right)}}{\left(x + 7\right)^{6} \left(3 x - 4\right)^{2} \sin^{3}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 5 \ln{\left(2 - 4 x \right)} + 7 \ln{\left(3 x + 5 \right)} + 8 \ln{\left(\cos{\left(x \right)} \right)}- x - 6 \ln{\left(x + 7 \right)} - 2 \ln{\left(3 x - 4 \right)} - 3 \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{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{21}{3 x + 5} - \frac{6}{3 x - 4} - \frac{6}{x + 7} - \frac{20}{2 - 4 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{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{21}{3 x + 5} - \frac{6}{3 x - 4} - \frac{6}{x + 7} - \frac{20}{2 - 4 x}\right)\left(\frac{\left(2 - 4 x\right)^{5} \left(3 x + 5\right)^{7} e^{- x} \cos^{8}{\left(x \right)}}{\left(x + 7\right)^{6} \left(3 x - 4\right)^{2} \sin^{3}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 8 \tan{\left(x \right)} + \frac{21}{3 x + 5} - \frac{20}{2 - 4 x}-1 - \frac{3}{\tan{\left(x \right)}} - \frac{6}{3 x - 4} - \frac{6}{x + 7}\right)\left(\frac{\left(2 - 4 x\right)^{5} \left(3 x + 5\right)^{7} e^{- x} \cos^{8}{\left(x \right)}}{\left(x + 7\right)^{6} \left(3 x - 4\right)^{2} \sin^{3}{\left(x \right)}} \right)