Find the derivative of LaTeX: \displaystyle y = \frac{\sqrt{5 x + 1} e^{- x} \sin^{2}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(3 - 8 x\right)^{6} \left(- 6 x - 3\right)^{3} \left(2 x - 6\right)^{7}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\sqrt{5 x + 1} e^{- x} \sin^{2}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(3 - 8 x\right)^{6} \left(- 6 x - 3\right)^{3} \left(2 x - 6\right)^{7}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{\ln{\left(5 x + 1 \right)}}{2} + 2 \ln{\left(\sin{\left(x \right)} \right)} + 7 \ln{\left(\cos{\left(x \right)} \right)}- x - 6 \ln{\left(3 - 8 x \right)} - 3 \ln{\left(- 6 x - 3 \right)} - 7 \ln{\left(2 x - 6 \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{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{5}{2 \left(5 x + 1\right)} - \frac{14}{2 x - 6} + \frac{18}{- 6 x - 3} + \frac{48}{3 - 8 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{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{5}{2 \left(5 x + 1\right)} - \frac{14}{2 x - 6} + \frac{18}{- 6 x - 3} + \frac{48}{3 - 8 x}\right)\left(\frac{\sqrt{5 x + 1} e^{- x} \sin^{2}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(3 - 8 x\right)^{6} \left(- 6 x - 3\right)^{3} \left(2 x - 6\right)^{7}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 7 \tan{\left(x \right)} + \frac{2}{\tan{\left(x \right)}} + \frac{5}{2 \left(5 x + 1\right)}-1 - \frac{14}{2 x - 6} + \frac{18}{- 6 x - 3} + \frac{48}{3 - 8 x}\right)\left(\frac{\sqrt{5 x + 1} e^{- x} \sin^{2}{\left(x \right)} \cos^{7}{\left(x \right)}}{\left(3 - 8 x\right)^{6} \left(- 6 x - 3\right)^{3} \left(2 x - 6\right)^{7}} \right)