Find the derivative of LaTeX: \displaystyle y = \frac{\left(9 - 3 x\right)^{4} \left(x + 8\right)^{5} \left(6 x + 1\right)^{8} e^{x}}{\sqrt{\left(9 x + 7\right)^{7}} \sin^{6}{\left(x \right)} \cos^{7}{\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 - 3 x\right)^{4} \left(x + 8\right)^{5} \left(6 x + 1\right)^{8} e^{x}}{\sqrt{\left(9 x + 7\right)^{7}} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 4 \ln{\left(9 - 3 x \right)} + 5 \ln{\left(x + 8 \right)} + 8 \ln{\left(6 x + 1 \right)}- \frac{7 \ln{\left(9 x + 7 \right)}}{2} - 6 \ln{\left(\sin{\left(x \right)} \right)} - 7 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{63}{2 \left(9 x + 7\right)} + \frac{48}{6 x + 1} + \frac{5}{x + 8} - \frac{12}{9 - 3 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{63}{2 \left(9 x + 7\right)} + \frac{48}{6 x + 1} + \frac{5}{x + 8} - \frac{12}{9 - 3 x}\right)\left(\frac{\left(9 - 3 x\right)^{4} \left(x + 8\right)^{5} \left(6 x + 1\right)^{8} e^{x}}{\sqrt{\left(9 x + 7\right)^{7}} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{48}{6 x + 1} + \frac{5}{x + 8} - \frac{12}{9 - 3 x}7 \tan{\left(x \right)} - \frac{6}{\tan{\left(x \right)}} - \frac{63}{2 \left(9 x + 7\right)}\right)\left(\frac{\left(9 - 3 x\right)^{4} \left(x + 8\right)^{5} \left(6 x + 1\right)^{8} e^{x}}{\sqrt{\left(9 x + 7\right)^{7}} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}} \right)