Find the derivative of LaTeX: \displaystyle y = \frac{\left(5 - 4 x\right)^{2} \left(3 x - 5\right)^{5} e^{x} \cos^{5}{\left(x \right)}}{\left(x - 4\right)^{8} \sqrt{\left(5 x + 7\right)^{7}} \sin^{8}{\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(5 - 4 x\right)^{2} \left(3 x - 5\right)^{5} e^{x} \cos^{5}{\left(x \right)}}{\left(x - 4\right)^{8} \sqrt{\left(5 x + 7\right)^{7}} \sin^{8}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 2 \ln{\left(5 - 4 x \right)} + 5 \ln{\left(3 x - 5 \right)} + 5 \ln{\left(\cos{\left(x \right)} \right)}- 8 \ln{\left(x - 4 \right)} - \frac{7 \ln{\left(5 x + 7 \right)}}{2} - 8 \ln{\left(\sin{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{35}{2 \left(5 x + 7\right)} + \frac{15}{3 x - 5} - \frac{8}{x - 4} - \frac{8}{5 - 4 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 - \frac{8 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{35}{2 \left(5 x + 7\right)} + \frac{15}{3 x - 5} - \frac{8}{x - 4} - \frac{8}{5 - 4 x}\right)\left(\frac{\left(5 - 4 x\right)^{2} \left(3 x - 5\right)^{5} e^{x} \cos^{5}{\left(x \right)}}{\left(x - 4\right)^{8} \sqrt{\left(5 x + 7\right)^{7}} \sin^{8}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 5 \tan{\left(x \right)} + 1 + \frac{15}{3 x - 5} - \frac{8}{5 - 4 x}- \frac{8}{\tan{\left(x \right)}} - \frac{35}{2 \left(5 x + 7\right)} - \frac{8}{x - 4}\right)\left(\frac{\left(5 - 4 x\right)^{2} \left(3 x - 5\right)^{5} e^{x} \cos^{5}{\left(x \right)}}{\left(x - 4\right)^{8} \sqrt{\left(5 x + 7\right)^{7}} \sin^{8}{\left(x \right)}} \right)