Find the derivative of LaTeX: \displaystyle y = \frac{\sqrt{3 x + 6} \left(5 x + 6\right)^{5} e^{- x} \sin^{6}{\left(x \right)}}{\left(4 - 7 x\right)^{5} \left(x - 8\right)^{6} \cos^{3}{\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{\sqrt{3 x + 6} \left(5 x + 6\right)^{5} e^{- x} \sin^{6}{\left(x \right)}}{\left(4 - 7 x\right)^{5} \left(x - 8\right)^{6} \cos^{3}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{\ln{\left(3 x + 6 \right)}}{2} + 5 \ln{\left(5 x + 6 \right)} + 6 \ln{\left(\sin{\left(x \right)} \right)}- x - 5 \ln{\left(4 - 7 x \right)} - 6 \ln{\left(x - 8 \right)} - 3 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{25}{5 x + 6} + \frac{3}{2 \left(3 x + 6\right)} - \frac{6}{x - 8} + \frac{35}{4 - 7 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{25}{5 x + 6} + \frac{3}{2 \left(3 x + 6\right)} - \frac{6}{x - 8} + \frac{35}{4 - 7 x}\right)\left(\frac{\sqrt{3 x + 6} \left(5 x + 6\right)^{5} e^{- x} \sin^{6}{\left(x \right)}}{\left(4 - 7 x\right)^{5} \left(x - 8\right)^{6} \cos^{3}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{6}{\tan{\left(x \right)}} + \frac{25}{5 x + 6} + \frac{3}{2 \left(3 x + 6\right)}3 \tan{\left(x \right)} - 1 - \frac{6}{x - 8} + \frac{35}{4 - 7 x}\right)\left(\frac{\sqrt{3 x + 6} \left(5 x + 6\right)^{5} e^{- x} \sin^{6}{\left(x \right)}}{\left(4 - 7 x\right)^{5} \left(x - 8\right)^{6} \cos^{3}{\left(x \right)}} \right)