Find the derivative of LaTeX: \displaystyle y = \frac{\left(- x - 8\right)^{5} \left(x + 3\right)^{3} e^{x} \cos^{7}{\left(x \right)}}{4782969 x^{7} \sqrt{\left(3 x + 1\right)^{3}}}

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 - 8\right)^{5} \left(x + 3\right)^{3} e^{x} \cos^{7}{\left(x \right)}}{4782969 x^{7} \sqrt{\left(3 x + 1\right)^{3}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = x + 5 \ln{\left(- x - 8 \right)} + 3 \ln{\left(x + 3 \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- 7 \ln{\left(x \right)} - \frac{3 \ln{\left(3 x + 1 \right)}}{2} - 14 \ln{\left(3 \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{9}{2 \left(3 x + 1\right)} + \frac{3}{x + 3} - \frac{5}{- x - 8} - \frac{7}{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{9}{2 \left(3 x + 1\right)} + \frac{3}{x + 3} - \frac{5}{- x - 8} - \frac{7}{x}\right)\left(\frac{\left(- x - 8\right)^{5} \left(x + 3\right)^{3} e^{x} \cos^{7}{\left(x \right)}}{4782969 x^{7} \sqrt{\left(3 x + 1\right)^{3}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 7 \tan{\left(x \right)} + 1 + \frac{3}{x + 3} - \frac{5}{- x - 8}- \frac{9}{2 \left(3 x + 1\right)} - \frac{7}{x}\right)\left(\frac{\left(- x - 8\right)^{5} \left(x + 3\right)^{3} e^{x} \cos^{7}{\left(x \right)}}{4782969 x^{7} \sqrt{\left(3 x + 1\right)^{3}}} \right)