Find the derivative of LaTeX: \displaystyle y = \frac{\left(3 - 3 x\right)^{5} \left(7 - 4 x\right)^{4} e^{x} \sin^{3}{\left(x \right)}}{\left(x - 9\right)^{7} \left(x + 8\right)^{2} \sqrt{\left(9 x + 5\right)^{7}}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(3 - 3 x\right)^{5} \left(7 - 4 x\right)^{4} e^{x} \sin^{3}{\left(x \right)}}{\left(x - 9\right)^{7} \left(x + 8\right)^{2} \sqrt{\left(9 x + 5\right)^{7}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 5 \ln{\left(3 - 3 x \right)} + 4 \ln{\left(7 - 4 x \right)} + 3 \ln{\left(\sin{\left(x \right)} \right)}- 7 \ln{\left(x - 9 \right)} - 2 \ln{\left(x + 8 \right)} - \frac{7 \ln{\left(9 x + 5 \right)}}{2} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{63}{2 \left(9 x + 5\right)} - \frac{2}{x + 8} - \frac{7}{x - 9} - \frac{16}{7 - 4 x} - \frac{15}{3 - 3 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{63}{2 \left(9 x + 5\right)} - \frac{2}{x + 8} - \frac{7}{x - 9} - \frac{16}{7 - 4 x} - \frac{15}{3 - 3 x}\right)\left(\frac{\left(3 - 3 x\right)^{5} \left(7 - 4 x\right)^{4} e^{x} \sin^{3}{\left(x \right)}}{\left(x - 9\right)^{7} \left(x + 8\right)^{2} \sqrt{\left(9 x + 5\right)^{7}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{3}{\tan{\left(x \right)}} - \frac{16}{7 - 4 x} - \frac{15}{3 - 3 x}- \frac{63}{2 \left(9 x + 5\right)} - \frac{2}{x + 8} - \frac{7}{x - 9}\right)\left(\frac{\left(3 - 3 x\right)^{5} \left(7 - 4 x\right)^{4} e^{x} \sin^{3}{\left(x \right)}}{\left(x - 9\right)^{7} \left(x + 8\right)^{2} \sqrt{\left(9 x + 5\right)^{7}}} \right)