Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 6 x - 4\right)^{2} e^{x} \sin^{8}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(3 - x\right)^{7} \left(x - 6\right)^{2} \sqrt{\left(5 x + 5\right)^{5}}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\left(- 6 x - 4\right)^{2} e^{x} \sin^{8}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(3 - x\right)^{7} \left(x - 6\right)^{2} \sqrt{\left(5 x + 5\right)^{5}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 2 \ln{\left(- 6 x - 4 \right)} + 8 \ln{\left(\sin{\left(x \right)} \right)} + 3 \ln{\left(\cos{\left(x \right)} \right)}- 7 \ln{\left(3 - x \right)} - 2 \ln{\left(x - 6 \right)} - \frac{5 \ln{\left(5 x + 5 \right)}}{2} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{25}{2 \left(5 x + 5\right)} - \frac{2}{x - 6} - \frac{12}{- 6 x - 4} + \frac{7}{3 - x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{25}{2 \left(5 x + 5\right)} - \frac{2}{x - 6} - \frac{12}{- 6 x - 4} + \frac{7}{3 - x}\right)\left(\frac{\left(- 6 x - 4\right)^{2} e^{x} \sin^{8}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(3 - x\right)^{7} \left(x - 6\right)^{2} \sqrt{\left(5 x + 5\right)^{5}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 3 \tan{\left(x \right)} + 1 + \frac{8}{\tan{\left(x \right)}} - \frac{12}{- 6 x - 4}- \frac{25}{2 \left(5 x + 5\right)} - \frac{2}{x - 6} + \frac{7}{3 - x}\right)\left(\frac{\left(- 6 x - 4\right)^{2} e^{x} \sin^{8}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(3 - x\right)^{7} \left(x - 6\right)^{2} \sqrt{\left(5 x + 5\right)^{5}}} \right)