Find the derivative of LaTeX: \displaystyle y = \frac{\left(2 - 8 x\right)^{7} \left(- 6 x - 6\right)^{4} \left(7 x + 5\right)^{6} e^{x}}{\sin^{7}{\left(x \right)} \cos^{7}{\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(2 - 8 x\right)^{7} \left(- 6 x - 6\right)^{4} \left(7 x + 5\right)^{6} e^{x}}{\sin^{7}{\left(x \right)} \cos^{7}{\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(2 - 8 x \right)} + 4 \ln{\left(- 6 x - 6 \right)} + 6 \ln{\left(7 x + 5 \right)}- 7 \ln{\left(\sin{\left(x \right)} \right)} - 7 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{42}{7 x + 5} - \frac{24}{- 6 x - 6} - \frac{56}{2 - 8 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{7 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{7 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{42}{7 x + 5} - \frac{24}{- 6 x - 6} - \frac{56}{2 - 8 x}\right)\left(\frac{\left(2 - 8 x\right)^{7} \left(- 6 x - 6\right)^{4} \left(7 x + 5\right)^{6} e^{x}}{\sin^{7}{\left(x \right)} \cos^{7}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{42}{7 x + 5} - \frac{24}{- 6 x - 6} - \frac{56}{2 - 8 x}7 \tan{\left(x \right)} - \frac{7}{\tan{\left(x \right)}}\right)\left(\frac{\left(2 - 8 x\right)^{7} \left(- 6 x - 6\right)^{4} \left(7 x + 5\right)^{6} e^{x}}{\sin^{7}{\left(x \right)} \cos^{7}{\left(x \right)}} \right)