Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 4\right)^{3} \sin^{2}{\left(x \right)}}{\left(x - 5\right)^{7} \left(2 x - 2\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 - 4\right)^{3} \sin^{2}{\left(x \right)}}{\left(x - 5\right)^{7} \left(2 x - 2\right)^{4}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 3 \ln{\left(x - 4 \right)} + 2 \ln{\left(\sin{\left(x \right)} \right)}- 7 \ln{\left(x - 5 \right)} - 4 \ln{\left(2 x - 2 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8}{2 x - 2} + \frac{3}{x - 4} - \frac{7}{x - 5} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8}{2 x - 2} + \frac{3}{x - 4} - \frac{7}{x - 5}\right)\left(\frac{\left(x - 4\right)^{3} \sin^{2}{\left(x \right)}}{\left(x - 5\right)^{7} \left(2 x - 2\right)^{4}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{2}{\tan{\left(x \right)}} + \frac{3}{x - 4}- \frac{8}{2 x - 2} - \frac{7}{x - 5}\right)\left(\frac{\left(x - 4\right)^{3} \sin^{2}{\left(x \right)}}{\left(x - 5\right)^{7} \left(2 x - 2\right)^{4}} \right)