Find the derivative of LaTeX: \displaystyle y = \frac{\left(5 x - 7\right)^{5} \sqrt{\left(9 x + 1\right)^{5}} e^{- x} \sin^{2}{\left(x \right)}}{\left(x - 4\right)^{3} \left(2 x + 2\right)^{3} \cos^{5}{\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 - 7\right)^{5} \sqrt{\left(9 x + 1\right)^{5}} e^{- x} \sin^{2}{\left(x \right)}}{\left(x - 4\right)^{3} \left(2 x + 2\right)^{3} \cos^{5}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 5 \ln{\left(5 x - 7 \right)} + \frac{5 \ln{\left(9 x + 1 \right)}}{2} + 2 \ln{\left(\sin{\left(x \right)} \right)}- x - 3 \ln{\left(x - 4 \right)} - 3 \ln{\left(2 x + 2 \right)} - 5 \ln{\left(\cos{\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{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{45}{2 \left(9 x + 1\right)} + \frac{25}{5 x - 7} - \frac{6}{2 x + 2} - \frac{3}{x - 4} 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{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{45}{2 \left(9 x + 1\right)} + \frac{25}{5 x - 7} - \frac{6}{2 x + 2} - \frac{3}{x - 4}\right)\left(\frac{\left(5 x - 7\right)^{5} \sqrt{\left(9 x + 1\right)^{5}} e^{- x} \sin^{2}{\left(x \right)}}{\left(x - 4\right)^{3} \left(2 x + 2\right)^{3} \cos^{5}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{2}{\tan{\left(x \right)}} + \frac{45}{2 \left(9 x + 1\right)} + \frac{25}{5 x - 7}5 \tan{\left(x \right)} - 1 - \frac{6}{2 x + 2} - \frac{3}{x - 4}\right)\left(\frac{\left(5 x - 7\right)^{5} \sqrt{\left(9 x + 1\right)^{5}} e^{- x} \sin^{2}{\left(x \right)}}{\left(x - 4\right)^{3} \left(2 x + 2\right)^{3} \cos^{5}{\left(x \right)}} \right)