Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 7\right)^{3} \left(7 x - 7\right)^{7} \sqrt{\left(9 x + 3\right)^{3}} e^{- x}}{\left(3 x - 7\right)^{6} \sin^{8}{\left(x \right)} \cos^{4}{\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)^{3} \left(7 x - 7\right)^{7} \sqrt{\left(9 x + 3\right)^{3}} e^{- x}}{\left(3 x - 7\right)^{6} \sin^{8}{\left(x \right)} \cos^{4}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 3 \ln{\left(x - 7 \right)} + 7 \ln{\left(7 x - 7 \right)} + \frac{3 \ln{\left(9 x + 3 \right)}}{2}- x - 6 \ln{\left(3 x - 7 \right)} - 8 \ln{\left(\sin{\left(x \right)} \right)} - 4 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{27}{2 \left(9 x + 3\right)} + \frac{49}{7 x - 7} - \frac{18}{3 x - 7} + \frac{3}{x - 7} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{27}{2 \left(9 x + 3\right)} + \frac{49}{7 x - 7} - \frac{18}{3 x - 7} + \frac{3}{x - 7}\right)\left(\frac{\left(x - 7\right)^{3} \left(7 x - 7\right)^{7} \sqrt{\left(9 x + 3\right)^{3}} e^{- x}}{\left(3 x - 7\right)^{6} \sin^{8}{\left(x \right)} \cos^{4}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{27}{2 \left(9 x + 3\right)} + \frac{49}{7 x - 7} + \frac{3}{x - 7}4 \tan{\left(x \right)} - 1 - \frac{8}{\tan{\left(x \right)}} - \frac{18}{3 x - 7}\right)\left(\frac{\left(x - 7\right)^{3} \left(7 x - 7\right)^{7} \sqrt{\left(9 x + 3\right)^{3}} e^{- x}}{\left(3 x - 7\right)^{6} \sin^{8}{\left(x \right)} \cos^{4}{\left(x \right)}} \right)