Find the derivative of LaTeX: \displaystyle y = \frac{\left(5 x - 8\right)^{4} \sqrt{\left(9 x + 2\right)^{5}} e^{- x} \sin^{4}{\left(x \right)}}{\left(x - 1\right)^{6} \left(5 x + 3\right)^{5} \cos^{6}{\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(5 x - 8\right)^{4} \sqrt{\left(9 x + 2\right)^{5}} e^{- x} \sin^{4}{\left(x \right)}}{\left(x - 1\right)^{6} \left(5 x + 3\right)^{5} \cos^{6}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 4 \ln{\left(5 x - 8 \right)} + \frac{5 \ln{\left(9 x + 2 \right)}}{2} + 4 \ln{\left(\sin{\left(x \right)} \right)}- x - 6 \ln{\left(x - 1 \right)} - 5 \ln{\left(5 x + 3 \right)} - 6 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{45}{2 \left(9 x + 2\right)} - \frac{25}{5 x + 3} + \frac{20}{5 x - 8} - \frac{6}{x - 1} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{45}{2 \left(9 x + 2\right)} - \frac{25}{5 x + 3} + \frac{20}{5 x - 8} - \frac{6}{x - 1}\right)\left(\frac{\left(5 x - 8\right)^{4} \sqrt{\left(9 x + 2\right)^{5}} e^{- x} \sin^{4}{\left(x \right)}}{\left(x - 1\right)^{6} \left(5 x + 3\right)^{5} \cos^{6}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{4}{\tan{\left(x \right)}} + \frac{45}{2 \left(9 x + 2\right)} + \frac{20}{5 x - 8}6 \tan{\left(x \right)} - 1 - \frac{25}{5 x + 3} - \frac{6}{x - 1}\right)\left(\frac{\left(5 x - 8\right)^{4} \sqrt{\left(9 x + 2\right)^{5}} e^{- x} \sin^{4}{\left(x \right)}}{\left(x - 1\right)^{6} \left(5 x + 3\right)^{5} \cos^{6}{\left(x \right)}} \right)