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

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{x^{7} e^{- x} \sin^{7}{\left(x \right)} \cos^{5}{\left(x \right)}}{\left(7 - 9 x\right)^{7} \left(8 x - 2\right)^{7} \sqrt{\left(7 x + 3\right)^{3}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 7 \ln{\left(x \right)} + 7 \ln{\left(\sin{\left(x \right)} \right)} + 5 \ln{\left(\cos{\left(x \right)} \right)}- x - 7 \ln{\left(7 - 9 x \right)} - \frac{3 \ln{\left(7 x + 3 \right)}}{2} - 7 \ln{\left(8 x - 2 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{56}{8 x - 2} - \frac{21}{2 \left(7 x + 3\right)} + \frac{63}{7 - 9 x} + \frac{7}{x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{56}{8 x - 2} - \frac{21}{2 \left(7 x + 3\right)} + \frac{63}{7 - 9 x} + \frac{7}{x}\right)\left(\frac{x^{7} e^{- x} \sin^{7}{\left(x \right)} \cos^{5}{\left(x \right)}}{\left(7 - 9 x\right)^{7} \left(8 x - 2\right)^{7} \sqrt{\left(7 x + 3\right)^{3}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 5 \tan{\left(x \right)} + \frac{7}{\tan{\left(x \right)}} + \frac{7}{x}-1 - \frac{56}{8 x - 2} - \frac{21}{2 \left(7 x + 3\right)} + \frac{63}{7 - 9 x}\right)\left(\frac{x^{7} e^{- x} \sin^{7}{\left(x \right)} \cos^{5}{\left(x \right)}}{\left(7 - 9 x\right)^{7} \left(8 x - 2\right)^{7} \sqrt{\left(7 x + 3\right)^{3}}} \right)