Find the derivative of LaTeX: \displaystyle y = \frac{59049 x^{5} \sqrt{9 x + 1}}{\left(4 - 7 x\right)^{8} \cos^{4}{\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{59049 x^{5} \sqrt{9 x + 1}}{\left(4 - 7 x\right)^{8} \cos^{4}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 5 \ln{\left(x \right)} + \frac{\ln{\left(9 x + 1 \right)}}{2} + 10 \ln{\left(3 \right)}- 8 \ln{\left(4 - 7 x \right)} - 4 \ln{\left(\cos{\left(x \right)} \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{9}{2 \left(9 x + 1\right)} + \frac{56}{4 - 7 x} + \frac{5}{x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{9}{2 \left(9 x + 1\right)} + \frac{56}{4 - 7 x} + \frac{5}{x}\right)\left(\frac{59049 x^{5} \sqrt{9 x + 1}}{\left(4 - 7 x\right)^{8} \cos^{4}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{9}{2 \left(9 x + 1\right)} + \frac{5}{x}4 \tan{\left(x \right)} + \frac{56}{4 - 7 x}\right)\left(\frac{59049 x^{5} \sqrt{9 x + 1}}{\left(4 - 7 x\right)^{8} \cos^{4}{\left(x \right)}} \right)