Find the derivative of LaTeX: \displaystyle y = \frac{\left(7 - 3 x\right)^{4} \sqrt{\left(3 x + 7\right)^{3}} e^{x}}{\left(2 x - 2\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(7 - 3 x\right)^{4} \sqrt{\left(3 x + 7\right)^{3}} e^{x}}{\left(2 x - 2\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 + 4 \ln{\left(7 - 3 x \right)} + \frac{3 \ln{\left(3 x + 7 \right)}}{2}- 2 \ln{\left(2 x - 2 \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{9}{2 \left(3 x + 7\right)} - \frac{4}{2 x - 2} - \frac{12}{7 - 3 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{9}{2 \left(3 x + 7\right)} - \frac{4}{2 x - 2} - \frac{12}{7 - 3 x}\right)\left(\frac{\left(7 - 3 x\right)^{4} \sqrt{\left(3 x + 7\right)^{3}} e^{x}}{\left(2 x - 2\right)^{2} \cos^{6}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{9}{2 \left(3 x + 7\right)} - \frac{12}{7 - 3 x}6 \tan{\left(x \right)} - \frac{4}{2 x - 2}\right)\left(\frac{\left(7 - 3 x\right)^{4} \sqrt{\left(3 x + 7\right)^{3}} e^{x}}{\left(2 x - 2\right)^{2} \cos^{6}{\left(x \right)}} \right)