Find the derivative of LaTeX: \displaystyle y = \frac{\left(9 - 2 x\right)^{2} \sin^{4}{\left(x \right)}}{\left(x - 2\right)^{6} \sqrt{7 x + 7}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(9 - 2 x\right)^{2} \sin^{4}{\left(x \right)}}{\left(x - 2\right)^{6} \sqrt{7 x + 7}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 2 \ln{\left(9 - 2 x \right)} + 4 \ln{\left(\sin{\left(x \right)} \right)}- 6 \ln{\left(x - 2 \right)} - \frac{\ln{\left(7 x + 7 \right)}}{2} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{7}{2 \left(7 x + 7\right)} - \frac{6}{x - 2} - \frac{4}{9 - 2 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{7}{2 \left(7 x + 7\right)} - \frac{6}{x - 2} - \frac{4}{9 - 2 x}\right)\left(\frac{\left(9 - 2 x\right)^{2} \sin^{4}{\left(x \right)}}{\left(x - 2\right)^{6} \sqrt{7 x + 7}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{4}{\tan{\left(x \right)}} - \frac{4}{9 - 2 x}- \frac{7}{2 \left(7 x + 7\right)} - \frac{6}{x - 2}\right)\left(\frac{\left(9 - 2 x\right)^{2} \sin^{4}{\left(x \right)}}{\left(x - 2\right)^{6} \sqrt{7 x + 7}} \right)