Find the derivative of LaTeX: \displaystyle y = \frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}}

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 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 5 \ln{\left(5 x + 6 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 2 \ln{\left(x \right)} - 8 \ln{\left(5 x - 2 \right)} - 2 \ln{\left(7 \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{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{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{25}{5 x + 6} - \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{25}{5 x + 6}- \frac{40}{5 x - 2} - \frac{2}{x}\right)\left(\frac{\left(5 x + 6\right)^{5} e^{x} \cos^{7}{\left(x \right)}}{49 x^{2} \left(5 x - 2\right)^{8}} \right)