Find the derivative of LaTeX: \displaystyle y = \frac{\left(6 x + 4\right)^{2} \left(7 x - 6\right)^{7} e^{x}}{x^{6} \left(6 x + 6\right)^{7} \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(6 x + 4\right)^{2} \left(7 x - 6\right)^{7} e^{x}}{x^{6} \left(6 x + 6\right)^{7} \sin^{2}{\left(x \right)}} \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)} + 7 \ln{\left(7 x - 6 \right)}- 6 \ln{\left(x \right)} - 7 \ln{\left(6 x + 6 \right)} - 2 \ln{\left(\sin{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = 1 - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{49}{7 x - 6} - \frac{42}{6 x + 6} + \frac{12}{6 x + 4} - \frac{6}{x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(1 - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{49}{7 x - 6} - \frac{42}{6 x + 6} + \frac{12}{6 x + 4} - \frac{6}{x}\right)\left(\frac{\left(6 x + 4\right)^{2} \left(7 x - 6\right)^{7} e^{x}}{x^{6} \left(6 x + 6\right)^{7} \sin^{2}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{49}{7 x - 6} + \frac{12}{6 x + 4}- \frac{2}{\tan{\left(x \right)}} - \frac{42}{6 x + 6} - \frac{6}{x}\right)\left(\frac{\left(6 x + 4\right)^{2} \left(7 x - 6\right)^{7} e^{x}}{x^{6} \left(6 x + 6\right)^{7} \sin^{2}{\left(x \right)}} \right)