Find the derivative of LaTeX: \displaystyle y = - \frac{16384 x^{7} \left(x + 9\right)^{5} e^{- x} \sin^{3}{\left(x \right)}}{\left(2 x - 9\right)^{4} \left(4 x - 1\right)^{4} \sqrt{\left(6 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{16384 x^{7} \left(x + 9\right)^{5} e^{- x} \sin^{3}{\left(x \right)}}{\left(2 x - 9\right)^{4} \left(4 x - 1\right)^{4} \sqrt{\left(6 x + 5\right)^{7}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(x \right)} + 5 \ln{\left(x + 9 \right)} + 3 \ln{\left(\sin{\left(x \right)} \right)} + 14 \ln{\left(2 \right)} + i \pi- x - 4 \ln{\left(2 x - 9 \right)} - 4 \ln{\left(4 x - 1 \right)} - \frac{7 \ln{\left(6 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{21}{6 x + 5} - \frac{16}{4 x - 1} - \frac{8}{2 x - 9} + \frac{5}{x + 9} + \frac{7}{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{21}{6 x + 5} - \frac{16}{4 x - 1} - \frac{8}{2 x - 9} + \frac{5}{x + 9} + \frac{7}{x}\right)\left(- \frac{16384 x^{7} \left(x + 9\right)^{5} e^{- x} \sin^{3}{\left(x \right)}}{\left(2 x - 9\right)^{4} \left(4 x - 1\right)^{4} \sqrt{\left(6 x + 5\right)^{7}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{3}{\tan{\left(x \right)}} + \frac{5}{x + 9} + \frac{7}{x}-1 - \frac{21}{6 x + 5} - \frac{16}{4 x - 1} - \frac{8}{2 x - 9}\right)\left(- \frac{16384 x^{7} \left(x + 9\right)^{5} e^{- x} \sin^{3}{\left(x \right)}}{\left(2 x - 9\right)^{4} \left(4 x - 1\right)^{4} \sqrt{\left(6 x + 5\right)^{7}}} \right)