Find the derivative of LaTeX: \displaystyle y = \frac{\left(9 x - 6\right)^{4} \cos^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{2} \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{\left(9 x - 6\right)^{4} \cos^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{2} \sin^{5}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 4 \ln{\left(9 x - 6 \right)} + 6 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(3 - 8 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)}} - \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{36}{9 x - 6} + \frac{16}{3 - 8 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)}} - \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{36}{9 x - 6} + \frac{16}{3 - 8 x}\right)\left(\frac{\left(9 x - 6\right)^{4} \cos^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{2} \sin^{5}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 6 \tan{\left(x \right)} + \frac{36}{9 x - 6}- \frac{5}{\tan{\left(x \right)}} + \frac{16}{3 - 8 x}\right)\left(\frac{\left(9 x - 6\right)^{4} \cos^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{2} \sin^{5}{\left(x \right)}} \right)