Find the derivative of LaTeX: \displaystyle y = \frac{\sqrt{7 x + 3} e^{- x} \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)}}{\left(1 - 8 x\right)^{8} \left(x - 7\right)^{8} \left(3 x - 9\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{\sqrt{7 x + 3} e^{- x} \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)}}{\left(1 - 8 x\right)^{8} \left(x - 7\right)^{8} \left(3 x - 9\right)^{3}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{\ln{\left(7 x + 3 \right)}}{2} + 7 \ln{\left(\sin{\left(x \right)} \right)} + 4 \ln{\left(\cos{\left(x \right)} \right)}- x - 8 \ln{\left(1 - 8 x \right)} - 8 \ln{\left(x - 7 \right)} - 3 \ln{\left(3 x - 9 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{7}{2 \left(7 x + 3\right)} - \frac{9}{3 x - 9} - \frac{8}{x - 7} + \frac{64}{1 - 8 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{7}{2 \left(7 x + 3\right)} - \frac{9}{3 x - 9} - \frac{8}{x - 7} + \frac{64}{1 - 8 x}\right)\left(\frac{\sqrt{7 x + 3} e^{- x} \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)}}{\left(1 - 8 x\right)^{8} \left(x - 7\right)^{8} \left(3 x - 9\right)^{3}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 4 \tan{\left(x \right)} + \frac{7}{\tan{\left(x \right)}} + \frac{7}{2 \left(7 x + 3\right)}-1 - \frac{9}{3 x - 9} - \frac{8}{x - 7} + \frac{64}{1 - 8 x}\right)\left(\frac{\sqrt{7 x + 3} e^{- x} \sin^{7}{\left(x \right)} \cos^{4}{\left(x \right)}}{\left(1 - 8 x\right)^{8} \left(x - 7\right)^{8} \left(3 x - 9\right)^{3}} \right)