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