Find the derivative of LaTeX: \displaystyle y = \frac{\sqrt{\left(x + 9\right)^{3}} e^{- x} \cos^{8}{\left(x \right)}}{\left(4 x - 7\right)^{5} \left(9 x - 2\right)^{2}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\sqrt{\left(x + 9\right)^{3}} e^{- x} \cos^{8}{\left(x \right)}}{\left(4 x - 7\right)^{5} \left(9 x - 2\right)^{2}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{3 \ln{\left(x + 9 \right)}}{2} + 8 \ln{\left(\cos{\left(x \right)} \right)}- x - 5 \ln{\left(4 x - 7 \right)} - 2 \ln{\left(9 x - 2 \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{18}{9 x - 2} - \frac{20}{4 x - 7} + \frac{3}{2 \left(x + 9\right)} 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{18}{9 x - 2} - \frac{20}{4 x - 7} + \frac{3}{2 \left(x + 9\right)}\right)\left(\frac{\sqrt{\left(x + 9\right)^{3}} e^{- x} \cos^{8}{\left(x \right)}}{\left(4 x - 7\right)^{5} \left(9 x - 2\right)^{2}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 8 \tan{\left(x \right)} + \frac{3}{2 \left(x + 9\right)}-1 - \frac{18}{9 x - 2} - \frac{20}{4 x - 7}\right)\left(\frac{\sqrt{\left(x + 9\right)^{3}} e^{- x} \cos^{8}{\left(x \right)}}{\left(4 x - 7\right)^{5} \left(9 x - 2\right)^{2}} \right)