Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 5\right)^{6} \left(3 x - 7\right)^{3} e^{- x} \cos^{5}{\left(x \right)}}{\left(3 - 2 x\right)^{8} \left(- 8 x - 4\right)^{7} \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 + 5\right)^{6} \left(3 x - 7\right)^{3} e^{- x} \cos^{5}{\left(x \right)}}{\left(3 - 2 x\right)^{8} \left(- 8 x - 4\right)^{7} \sin^{7}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(x + 5 \right)} + 3 \ln{\left(3 x - 7 \right)} + 5 \ln{\left(\cos{\left(x \right)} \right)}- x - 8 \ln{\left(3 - 2 x \right)} - 7 \ln{\left(- 8 x - 4 \right)} - 7 \ln{\left(\sin{\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{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{9}{3 x - 7} + \frac{6}{x + 5} + \frac{56}{- 8 x - 4} + \frac{16}{3 - 2 x} 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{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{9}{3 x - 7} + \frac{6}{x + 5} + \frac{56}{- 8 x - 4} + \frac{16}{3 - 2 x}\right)\left(\frac{\left(x + 5\right)^{6} \left(3 x - 7\right)^{3} e^{- x} \cos^{5}{\left(x \right)}}{\left(3 - 2 x\right)^{8} \left(- 8 x - 4\right)^{7} \sin^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 5 \tan{\left(x \right)} + \frac{9}{3 x - 7} + \frac{6}{x + 5}-1 - \frac{7}{\tan{\left(x \right)}} + \frac{56}{- 8 x - 4} + \frac{16}{3 - 2 x}\right)\left(\frac{\left(x + 5\right)^{6} \left(3 x - 7\right)^{3} e^{- x} \cos^{5}{\left(x \right)}}{\left(3 - 2 x\right)^{8} \left(- 8 x - 4\right)^{7} \sin^{7}{\left(x \right)}} \right)