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

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 - 3\right)^{6} e^{- x} \cos^{4}{\left(x \right)}}{\left(x - 5\right)^{6} \left(3 x - 3\right)^{8} \left(5 x - 3\right)^{2}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(x - 3 \right)} + 4 \ln{\left(\cos{\left(x \right)} \right)}- x - 6 \ln{\left(x - 5 \right)} - 8 \ln{\left(3 x - 3 \right)} - 2 \ln{\left(5 x - 3 \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{10}{5 x - 3} - \frac{24}{3 x - 3} + \frac{6}{x - 3} - \frac{6}{x - 5} 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{10}{5 x - 3} - \frac{24}{3 x - 3} + \frac{6}{x - 3} - \frac{6}{x - 5}\right)\left(\frac{\left(x - 3\right)^{6} e^{- x} \cos^{4}{\left(x \right)}}{\left(x - 5\right)^{6} \left(3 x - 3\right)^{8} \left(5 x - 3\right)^{2}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 4 \tan{\left(x \right)} + \frac{6}{x - 3}-1 - \frac{10}{5 x - 3} - \frac{24}{3 x - 3} - \frac{6}{x - 5}\right)\left(\frac{\left(x - 3\right)^{6} e^{- x} \cos^{4}{\left(x \right)}}{\left(x - 5\right)^{6} \left(3 x - 3\right)^{8} \left(5 x - 3\right)^{2}} \right)