Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 7\right)^{4} \left(3 x - 7\right)^{2} \left(7 x - 3\right)^{6} e^{- x}}{\left(4 x - 3\right)^{7} \sqrt{\left(9 x + 2\right)^{5}} \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 - 7\right)^{4} \left(3 x - 7\right)^{2} \left(7 x - 3\right)^{6} e^{- x}}{\left(4 x - 3\right)^{7} \sqrt{\left(9 x + 2\right)^{5}} \cos^{7}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 4 \ln{\left(x - 7 \right)} + 2 \ln{\left(3 x - 7 \right)} + 6 \ln{\left(7 x - 3 \right)}- x - 7 \ln{\left(4 x - 3 \right)} - \frac{5 \ln{\left(9 x + 2 \right)}}{2} - 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{45}{2 \left(9 x + 2\right)} + \frac{42}{7 x - 3} - \frac{28}{4 x - 3} + \frac{6}{3 x - 7} + \frac{4}{x - 7} 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{45}{2 \left(9 x + 2\right)} + \frac{42}{7 x - 3} - \frac{28}{4 x - 3} + \frac{6}{3 x - 7} + \frac{4}{x - 7}\right)\left(\frac{\left(x - 7\right)^{4} \left(3 x - 7\right)^{2} \left(7 x - 3\right)^{6} e^{- x}}{\left(4 x - 3\right)^{7} \sqrt{\left(9 x + 2\right)^{5}} \cos^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{42}{7 x - 3} + \frac{6}{3 x - 7} + \frac{4}{x - 7}7 \tan{\left(x \right)} - 1 - \frac{45}{2 \left(9 x + 2\right)} - \frac{28}{4 x - 3}\right)\left(\frac{\left(x - 7\right)^{4} \left(3 x - 7\right)^{2} \left(7 x - 3\right)^{6} e^{- x}}{\left(4 x - 3\right)^{7} \sqrt{\left(9 x + 2\right)^{5}} \cos^{7}{\left(x \right)}} \right)