Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 9\right)^{8} \left(8 x + 7\right)^{7} e^{- x}}{\left(2 x + 7\right)^{8} \cos^{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 + 9\right)^{8} \left(8 x + 7\right)^{7} e^{- x}}{\left(2 x + 7\right)^{8} \cos^{7}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 8 \ln{\left(x + 9 \right)} + 7 \ln{\left(8 x + 7 \right)}- x - 8 \ln{\left(2 x + 7 \right)} - 7 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{56}{8 x + 7} - \frac{16}{2 x + 7} + \frac{8}{x + 9} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{56}{8 x + 7} - \frac{16}{2 x + 7} + \frac{8}{x + 9}\right)\left(\frac{\left(x + 9\right)^{8} \left(8 x + 7\right)^{7} e^{- x}}{\left(2 x + 7\right)^{8} \cos^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{56}{8 x + 7} + \frac{8}{x + 9}7 \tan{\left(x \right)} - 1 - \frac{16}{2 x + 7}\right)\left(\frac{\left(x + 9\right)^{8} \left(8 x + 7\right)^{7} e^{- x}}{\left(2 x + 7\right)^{8} \cos^{7}{\left(x \right)}} \right)