Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 2\right)^{4} \sqrt{\left(x + 2\right)^{3}}}{\left(4 x - 7\right)^{3} \sin^{8}{\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 - 2\right)^{4} \sqrt{\left(x + 2\right)^{3}}}{\left(4 x - 7\right)^{3} \sin^{8}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 4 \ln{\left(x - 2 \right)} + \frac{3 \ln{\left(x + 2 \right)}}{2}- 3 \ln{\left(4 x - 7 \right)} - 8 \ln{\left(\sin{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{12}{4 x - 7} + \frac{3}{2 \left(x + 2\right)} + \frac{4}{x - 2} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{12}{4 x - 7} + \frac{3}{2 \left(x + 2\right)} + \frac{4}{x - 2}\right)\left(\frac{\left(x - 2\right)^{4} \sqrt{\left(x + 2\right)^{3}}}{\left(4 x - 7\right)^{3} \sin^{8}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{3}{2 \left(x + 2\right)} + \frac{4}{x - 2}- \frac{8}{\tan{\left(x \right)}} - \frac{12}{4 x - 7}\right)\left(\frac{\left(x - 2\right)^{4} \sqrt{\left(x + 2\right)^{3}}}{\left(4 x - 7\right)^{3} \sin^{8}{\left(x \right)}} \right)