Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 2\right)^{6} e^{- x} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(6 - 2 x\right)^{3} \left(9 x - 8\right)^{4}}

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 + 2\right)^{6} e^{- x} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(6 - 2 x\right)^{3} \left(9 x - 8\right)^{4}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(x + 2 \right)} + 4 \ln{\left(\sin{\left(x \right)} \right)} + 2 \ln{\left(\cos{\left(x \right)} \right)}- x - 3 \ln{\left(6 - 2 x \right)} - 4 \ln{\left(9 x - 8 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{36}{9 x - 8} + \frac{6}{x + 2} + \frac{6}{6 - 2 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{36}{9 x - 8} + \frac{6}{x + 2} + \frac{6}{6 - 2 x}\right)\left(\frac{\left(x + 2\right)^{6} e^{- x} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(6 - 2 x\right)^{3} \left(9 x - 8\right)^{4}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 2 \tan{\left(x \right)} + \frac{4}{\tan{\left(x \right)}} + \frac{6}{x + 2}-1 - \frac{36}{9 x - 8} + \frac{6}{6 - 2 x}\right)\left(\frac{\left(x + 2\right)^{6} e^{- x} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(6 - 2 x\right)^{3} \left(9 x - 8\right)^{4}} \right)