Find the derivative of LaTeX: \displaystyle y = \frac{\left(5 x - 7\right)^{6} e^{- x} \cos^{7}{\left(x \right)}}{\left(x - 6\right)^{6} \sqrt{6 x + 5} \left(8 x - 1\right)^{7}}

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)^{6} e^{- x} \cos^{7}{\left(x \right)}}{\left(x - 6\right)^{6} \sqrt{6 x + 5} \left(8 x - 1\right)^{7}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(5 x - 7 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- x - 6 \ln{\left(x - 6 \right)} - \frac{\ln{\left(6 x + 5 \right)}}{2} - 7 \ln{\left(8 x - 1 \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 - 1} - \frac{3}{6 x + 5} + \frac{30}{5 x - 7} - \frac{6}{x - 6} 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 - 1} - \frac{3}{6 x + 5} + \frac{30}{5 x - 7} - \frac{6}{x - 6}\right)\left(\frac{\left(5 x - 7\right)^{6} e^{- x} \cos^{7}{\left(x \right)}}{\left(x - 6\right)^{6} \sqrt{6 x + 5} \left(8 x - 1\right)^{7}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 7 \tan{\left(x \right)} + \frac{30}{5 x - 7}-1 - \frac{56}{8 x - 1} - \frac{3}{6 x + 5} - \frac{6}{x - 6}\right)\left(\frac{\left(5 x - 7\right)^{6} e^{- x} \cos^{7}{\left(x \right)}}{\left(x - 6\right)^{6} \sqrt{6 x + 5} \left(8 x - 1\right)^{7}} \right)