Find the derivative of LaTeX: \displaystyle y = \frac{\left(x - 5\right)^{6} \sqrt{6 x + 1}}{7776 x^{5} \left(- x - 5\right)^{2} \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{\left(x - 5\right)^{6} \sqrt{6 x + 1}}{7776 x^{5} \left(- x - 5\right)^{2} \cos^{4}{\left(x \right)}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 6 \ln{\left(x - 5 \right)} + \frac{\ln{\left(6 x + 1 \right)}}{2}- 5 \ln{\left(x \right)} - 2 \ln{\left(- x - 5 \right)} - 4 \ln{\left(\cos{\left(x \right)} \right)} - 5 \ln{\left(6 \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{3}{6 x + 1} + \frac{6}{x - 5} + \frac{2}{- x - 5} - \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{3}{6 x + 1} + \frac{6}{x - 5} + \frac{2}{- x - 5} - \frac{5}{x}\right)\left(\frac{\left(x - 5\right)^{6} \sqrt{6 x + 1}}{7776 x^{5} \left(- x - 5\right)^{2} \cos^{4}{\left(x \right)}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(\frac{3}{6 x + 1} + \frac{6}{x - 5}4 \tan{\left(x \right)} + \frac{2}{- x - 5} - \frac{5}{x}\right)\left(\frac{\left(x - 5\right)^{6} \sqrt{6 x + 1}}{7776 x^{5} \left(- x - 5\right)^{2} \cos^{4}{\left(x \right)}} \right)