Find the derivative of LaTeX: \displaystyle y = \frac{\left(- 3 x - 7\right)^{7} \sqrt{\left(9 x + 7\right)^{3}} e^{x} \sin^{3}{\left(x \right)}}{\left(3 - 2 x\right)^{2} \cos^{6}{\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 x - 7\right)^{7} \sqrt{\left(9 x + 7\right)^{3}} e^{x} \sin^{3}{\left(x \right)}}{\left(3 - 2 x\right)^{2} \cos^{6}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 7 \ln{\left(- 3 x - 7 \right)} + \frac{3 \ln{\left(9 x + 7 \right)}}{2} + 3 \ln{\left(\sin{\left(x \right)} \right)}- 2 \ln{\left(3 - 2 x \right)} - 6 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{27}{2 \left(9 x + 7\right)} - \frac{21}{- 3 x - 7} + \frac{4}{3 - 2 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{27}{2 \left(9 x + 7\right)} - \frac{21}{- 3 x - 7} + \frac{4}{3 - 2 x}\right)\left(\frac{\left(- 3 x - 7\right)^{7} \sqrt{\left(9 x + 7\right)^{3}} e^{x} \sin^{3}{\left(x \right)}}{\left(3 - 2 x\right)^{2} \cos^{6}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{3}{\tan{\left(x \right)}} + \frac{27}{2 \left(9 x + 7\right)} - \frac{21}{- 3 x - 7}6 \tan{\left(x \right)} + \frac{4}{3 - 2 x}\right)\left(\frac{\left(- 3 x - 7\right)^{7} \sqrt{\left(9 x + 7\right)^{3}} e^{x} \sin^{3}{\left(x \right)}}{\left(3 - 2 x\right)^{2} \cos^{6}{\left(x \right)}} \right)