Find the derivative of LaTeX: \displaystyle y = \frac{\left(3 - 9 x\right)^{4} \left(6 x + 6\right)^{2} e^{x} \sin^{2}{\left(x \right)}}{\left(x - 7\right)^{2} \sqrt{5 x + 6} \cos^{4}{\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(3 - 9 x\right)^{4} \left(6 x + 6\right)^{2} e^{x} \sin^{2}{\left(x \right)}}{\left(x - 7\right)^{2} \sqrt{5 x + 6} \cos^{4}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 4 \ln{\left(3 - 9 x \right)} + 2 \ln{\left(6 x + 6 \right)} + 2 \ln{\left(\sin{\left(x \right)} \right)}- 2 \ln{\left(x - 7 \right)} - \frac{\ln{\left(5 x + 6 \right)}}{2} - 4 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{12}{6 x + 6} - \frac{5}{2 \left(5 x + 6\right)} - \frac{2}{x - 7} - \frac{36}{3 - 9 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{12}{6 x + 6} - \frac{5}{2 \left(5 x + 6\right)} - \frac{2}{x - 7} - \frac{36}{3 - 9 x}\right)\left(\frac{\left(3 - 9 x\right)^{4} \left(6 x + 6\right)^{2} e^{x} \sin^{2}{\left(x \right)}}{\left(x - 7\right)^{2} \sqrt{5 x + 6} \cos^{4}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{2}{\tan{\left(x \right)}} + \frac{12}{6 x + 6} - \frac{36}{3 - 9 x}4 \tan{\left(x \right)} - \frac{5}{2 \left(5 x + 6\right)} - \frac{2}{x - 7}\right)\left(\frac{\left(3 - 9 x\right)^{4} \left(6 x + 6\right)^{2} e^{x} \sin^{2}{\left(x \right)}}{\left(x - 7\right)^{2} \sqrt{5 x + 6} \cos^{4}{\left(x \right)}} \right)