Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 4\right)^{5} e^{- x} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(x + 7\right)^{5} \left(4 x - 1\right)^{4} \left(7 x + 2\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{\left(x - 4\right)^{5} e^{- x} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(x + 7\right)^{5} \left(4 x - 1\right)^{4} \left(7 x + 2\right)^{3}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 5 \ln{\left(x - 4 \right)} + 6 \ln{\left(\sin{\left(x \right)} \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- x - 5 \ln{\left(x + 7 \right)} - 4 \ln{\left(4 x - 1 \right)} - 3 \ln{\left(7 x + 2 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{21}{7 x + 2} - \frac{16}{4 x - 1} - \frac{5}{x + 7} + \frac{5}{x - 4} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{21}{7 x + 2} - \frac{16}{4 x - 1} - \frac{5}{x + 7} + \frac{5}{x - 4}\right)\left(\frac{\left(x - 4\right)^{5} e^{- x} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(x + 7\right)^{5} \left(4 x - 1\right)^{4} \left(7 x + 2\right)^{3}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 7 \tan{\left(x \right)} + \frac{6}{\tan{\left(x \right)}} + \frac{5}{x - 4}-1 - \frac{21}{7 x + 2} - \frac{16}{4 x - 1} - \frac{5}{x + 7}\right)\left(\frac{\left(x - 4\right)^{5} e^{- x} \sin^{6}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(x + 7\right)^{5} \left(4 x - 1\right)^{4} \left(7 x + 2\right)^{3}} \right)