Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 7\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 7 x - 7\right)^{2} \left(7 x - 1\right)^{4}}

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 - 7\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 7 x - 7\right)^{2} \left(7 x - 1\right)^{4}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(x - 7 \right)} + 7 \ln{\left(\sin{\left(x \right)} \right)}- x - 2 \ln{\left(- 7 x - 7 \right)} - 4 \ln{\left(7 x - 1 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = -1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{28}{7 x - 1} + \frac{6}{x - 7} + \frac{14}{- 7 x - 7} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(-1 + \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{28}{7 x - 1} + \frac{6}{x - 7} + \frac{14}{- 7 x - 7}\right)\left(\frac{\left(x - 7\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 7 x - 7\right)^{2} \left(7 x - 1\right)^{4}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{7}{\tan{\left(x \right)}} + \frac{6}{x - 7}-1 - \frac{28}{7 x - 1} + \frac{14}{- 7 x - 7}\right)\left(\frac{\left(x - 7\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(- 7 x - 7\right)^{2} \left(7 x - 1\right)^{4}} \right)