Find the derivative of LaTeX: \displaystyle y = \frac{\sqrt{\left(6 x + 4\right)^{7}} \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(x - 1\right)^{4} \left(3 x - 4\right)^{8} \left(4 x - 7\right)^{3}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\sqrt{\left(6 x + 4\right)^{7}} \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(x - 1\right)^{4} \left(3 x - 4\right)^{8} \left(4 x - 7\right)^{3}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{7 \ln{\left(6 x + 4 \right)}}{2} + 3 \ln{\left(\sin{\left(x \right)} \right)} + 3 \ln{\left(\cos{\left(x \right)} \right)}- 4 \ln{\left(x - 1 \right)} - 8 \ln{\left(3 x - 4 \right)} - 3 \ln{\left(4 x - 7 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{21}{6 x + 4} - \frac{12}{4 x - 7} - \frac{24}{3 x - 4} - \frac{4}{x - 1} 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)}} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{21}{6 x + 4} - \frac{12}{4 x - 7} - \frac{24}{3 x - 4} - \frac{4}{x - 1}\right)\left(\frac{\sqrt{\left(6 x + 4\right)^{7}} \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(x - 1\right)^{4} \left(3 x - 4\right)^{8} \left(4 x - 7\right)^{3}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 3 \tan{\left(x \right)} + \frac{3}{\tan{\left(x \right)}} + \frac{21}{6 x + 4}- \frac{12}{4 x - 7} - \frac{24}{3 x - 4} - \frac{4}{x - 1}\right)\left(\frac{\sqrt{\left(6 x + 4\right)^{7}} \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{\left(x - 1\right)^{4} \left(3 x - 4\right)^{8} \left(4 x - 7\right)^{3}} \right)