Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 2\right)^{2} \sqrt{\left(9 x + 9\right)^{7}} e^{- x} \cos^{7}{\left(x \right)}}{\left(- 3 x - 5\right)^{4} \left(4 x - 6\right)^{2} \left(5 x + 3\right)^{8}}

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 + 2\right)^{2} \sqrt{\left(9 x + 9\right)^{7}} e^{- x} \cos^{7}{\left(x \right)}}{\left(- 3 x - 5\right)^{4} \left(4 x - 6\right)^{2} \left(5 x + 3\right)^{8}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 2 \ln{\left(x + 2 \right)} + \frac{7 \ln{\left(9 x + 9 \right)}}{2} + 7 \ln{\left(\cos{\left(x \right)} \right)}- x - 4 \ln{\left(- 3 x - 5 \right)} - 2 \ln{\left(4 x - 6 \right)} - 8 \ln{\left(5 x + 3 \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{63}{2 \left(9 x + 9\right)} - \frac{40}{5 x + 3} - \frac{8}{4 x - 6} + \frac{2}{x + 2} + \frac{12}{- 3 x - 5} 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{63}{2 \left(9 x + 9\right)} - \frac{40}{5 x + 3} - \frac{8}{4 x - 6} + \frac{2}{x + 2} + \frac{12}{- 3 x - 5}\right)\left(\frac{\left(x + 2\right)^{2} \sqrt{\left(9 x + 9\right)^{7}} e^{- x} \cos^{7}{\left(x \right)}}{\left(- 3 x - 5\right)^{4} \left(4 x - 6\right)^{2} \left(5 x + 3\right)^{8}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 7 \tan{\left(x \right)} + \frac{63}{2 \left(9 x + 9\right)} + \frac{2}{x + 2}-1 - \frac{40}{5 x + 3} - \frac{8}{4 x - 6} + \frac{12}{- 3 x - 5}\right)\left(\frac{\left(x + 2\right)^{2} \sqrt{\left(9 x + 9\right)^{7}} e^{- x} \cos^{7}{\left(x \right)}}{\left(- 3 x - 5\right)^{4} \left(4 x - 6\right)^{2} \left(5 x + 3\right)^{8}} \right)