Find the derivative of LaTeX: \displaystyle y = \frac{\left(7 x - 6\right)^{4} \sqrt{\left(5 x + 8\right)^{7}}}{\left(x - 3\right)^{2} \sin^{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(7 x - 6\right)^{4} \sqrt{\left(5 x + 8\right)^{7}}}{\left(x - 3\right)^{2} \sin^{2}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{7 \ln{\left(5 x + 8 \right)}}{2} + 4 \ln{\left(7 x - 6 \right)}- 2 \ln{\left(x - 3 \right)} - 2 \ln{\left(\sin{\left(x \right)} \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{28}{7 x - 6} + \frac{35}{2 \left(5 x + 8\right)} - \frac{2}{x - 3} 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{28}{7 x - 6} + \frac{35}{2 \left(5 x + 8\right)} - \frac{2}{x - 3}\right)\left(\frac{\left(7 x - 6\right)^{4} \sqrt{\left(5 x + 8\right)^{7}}}{\left(x - 3\right)^{2} \sin^{2}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{28}{7 x - 6} + \frac{35}{2 \left(5 x + 8\right)}- \frac{2}{\tan{\left(x \right)}} - \frac{2}{x - 3}\right)\left(\frac{\left(7 x - 6\right)^{4} \sqrt{\left(5 x + 8\right)^{7}}}{\left(x - 3\right)^{2} \sin^{2}{\left(x \right)}} \right)