Find the derivative of LaTeX: \displaystyle y = \frac{7776 x^{5} e^{- x} \cos^{6}{\left(x \right)}}{\left(3 - 2 x\right)^{5} \sin^{5}{\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{7776 x^{5} e^{- x} \cos^{6}{\left(x \right)}}{\left(3 - 2 x\right)^{5} \sin^{5}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 5 \ln{\left(x \right)} + 6 \ln{\left(\cos{\left(x \right)} \right)} + 5 \ln{\left(6 \right)}- x - 5 \ln{\left(3 - 2 x \right)} - 5 \ln{\left(\sin{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 - \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{10}{3 - 2 x} + \frac{5}{x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 - \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{10}{3 - 2 x} + \frac{5}{x}\right)\left(\frac{7776 x^{5} e^{- x} \cos^{6}{\left(x \right)}}{\left(3 - 2 x\right)^{5} \sin^{5}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 6 \tan{\left(x \right)} + \frac{5}{x}-1 - \frac{5}{\tan{\left(x \right)}} + \frac{10}{3 - 2 x}\right)\left(\frac{7776 x^{5} e^{- x} \cos^{6}{\left(x \right)}}{\left(3 - 2 x\right)^{5} \sin^{5}{\left(x \right)}} \right)