Find the derivative of LaTeX: \displaystyle y = \frac{\left(6 x + 6\right)^{6} e^{- x} \sin^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{4} \left(x + 6\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(6 x + 6\right)^{6} e^{- x} \sin^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{4} \left(x + 6\right)^{8}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(6 x + 6 \right)} + 6 \ln{\left(\sin{\left(x \right)} \right)}- x - 4 \ln{\left(3 - 8 x \right)} - 8 \ln{\left(x + 6 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{36}{6 x + 6} - \frac{8}{x + 6} + \frac{32}{3 - 8 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{36}{6 x + 6} - \frac{8}{x + 6} + \frac{32}{3 - 8 x}\right)\left(\frac{\left(6 x + 6\right)^{6} e^{- x} \sin^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{4} \left(x + 6\right)^{8}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{6}{\tan{\left(x \right)}} + \frac{36}{6 x + 6}-1 - \frac{8}{x + 6} + \frac{32}{3 - 8 x}\right)\left(\frac{\left(6 x + 6\right)^{6} e^{- x} \sin^{6}{\left(x \right)}}{\left(3 - 8 x\right)^{4} \left(x + 6\right)^{8}} \right)