Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 7\right)^{2} \left(7 x - 2\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(7 - 6 x\right)^{2} \left(- 5 x - 8\right)^{4} \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 - 7\right)^{2} \left(7 x - 2\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(7 - 6 x\right)^{2} \left(- 5 x - 8\right)^{4} \cos^{2}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 2 \ln{\left(x - 7 \right)} + 6 \ln{\left(7 x - 2 \right)} + 7 \ln{\left(\sin{\left(x \right)} \right)}- x - 2 \ln{\left(7 - 6 x \right)} - 4 \ln{\left(- 5 x - 8 \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{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{42}{7 x - 2} + \frac{2}{x - 7} + \frac{20}{- 5 x - 8} + \frac{12}{7 - 6 x} 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{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{42}{7 x - 2} + \frac{2}{x - 7} + \frac{20}{- 5 x - 8} + \frac{12}{7 - 6 x}\right)\left(\frac{\left(x - 7\right)^{2} \left(7 x - 2\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(7 - 6 x\right)^{2} \left(- 5 x - 8\right)^{4} \cos^{2}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{7}{\tan{\left(x \right)}} + \frac{42}{7 x - 2} + \frac{2}{x - 7}2 \tan{\left(x \right)} - 1 + \frac{20}{- 5 x - 8} + \frac{12}{7 - 6 x}\right)\left(\frac{\left(x - 7\right)^{2} \left(7 x - 2\right)^{6} e^{- x} \sin^{7}{\left(x \right)}}{\left(7 - 6 x\right)^{2} \left(- 5 x - 8\right)^{4} \cos^{2}{\left(x \right)}} \right)