Find the derivative of LaTeX: \displaystyle y = \frac{\left(x + 5\right)^{4} e^{- x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(6 - 6 x\right)^{2} \left(5 x + 5\right)^{3} \sqrt{\left(5 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{\left(x + 5\right)^{4} e^{- x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(6 - 6 x\right)^{2} \left(5 x + 5\right)^{3} \sqrt{\left(5 x + 6\right)^{7}}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = 4 \ln{\left(x + 5 \right)} + 2 \ln{\left(\sin{\left(x \right)} \right)} + 6 \ln{\left(\cos{\left(x \right)} \right)}- x - 2 \ln{\left(6 - 6 x \right)} - 3 \ln{\left(5 x + 5 \right)} - \frac{7 \ln{\left(5 x + 6 \right)}}{2} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = - \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{35}{2 \left(5 x + 6\right)} - \frac{15}{5 x + 5} + \frac{4}{x + 5} + \frac{12}{6 - 6 x} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(- \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}} - 1 + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{35}{2 \left(5 x + 6\right)} - \frac{15}{5 x + 5} + \frac{4}{x + 5} + \frac{12}{6 - 6 x}\right)\left(\frac{\left(x + 5\right)^{4} e^{- x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(6 - 6 x\right)^{2} \left(5 x + 5\right)^{3} \sqrt{\left(5 x + 6\right)^{7}}} \right) Using some Trigonometric identities to simplify gives LaTeX: y' = \left(- 6 \tan{\left(x \right)} + \frac{2}{\tan{\left(x \right)}} + \frac{4}{x + 5}-1 - \frac{35}{2 \left(5 x + 6\right)} - \frac{15}{5 x + 5} + \frac{12}{6 - 6 x}\right)\left(\frac{\left(x + 5\right)^{4} e^{- x} \sin^{2}{\left(x \right)} \cos^{6}{\left(x \right)}}{\left(6 - 6 x\right)^{2} \left(5 x + 5\right)^{3} \sqrt{\left(5 x + 6\right)^{7}}} \right)