Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 2\right)^{3} \left(5 x - 5\right)^{2} e^{x}}{\sqrt{\left(7 x + 4\right)^{3}} \cos^{5}{\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(x + 2\right)^{3} \left(5 x - 5\right)^{2} e^{x}}{\sqrt{\left(7 x + 4\right)^{3}} \cos^{5}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 3 \ln{\left(x + 2 \right)} + 2 \ln{\left(5 x - 5 \right)}- \frac{3 \ln{\left(7 x + 4 \right)}}{2} - 5 \ln{\left(\cos{\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{21}{2 \left(7 x + 4\right)} + \frac{10}{5 x - 5} + \frac{3}{x + 2} 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{21}{2 \left(7 x + 4\right)} + \frac{10}{5 x - 5} + \frac{3}{x + 2}\right)\left(\frac{\left(x + 2\right)^{3} \left(5 x - 5\right)^{2} e^{x}}{\sqrt{\left(7 x + 4\right)^{3}} \cos^{5}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(1 + \frac{10}{5 x - 5} + \frac{3}{x + 2}5 \tan{\left(x \right)} - \frac{21}{2 \left(7 x + 4\right)}\right)\left(\frac{\left(x + 2\right)^{3} \left(5 x - 5\right)^{2} e^{x}}{\sqrt{\left(7 x + 4\right)^{3}} \cos^{5}{\left(x \right)}} \right)