Find the derivative of LaTeX: \displaystyle y = \frac{\sqrt{\left(2 x + 6\right)^{3}}}{\left(x + 3\right)^{5}}

Taking the natural logarithm of both sides of the equation and expanding the right hand side gives: LaTeX: \ln(y) = \ln{\left(\frac{\sqrt{\left(2 x + 6\right)^{3}}}{\left(x + 3\right)^{5}} \right)} Expanding the right hand side using the product and quotient properties of logarithms gives: LaTeX: \ln(y) = \frac{3 \ln{\left(2 x + 6 \right)}}{2}- 5 \ln{\left(x + 3 \right)} Taking the derivative on both sides of the equation yields: LaTeX: \frac{y'}{y} = \frac{3}{2 x + 6} - \frac{5}{x + 3} Solving for LaTeX: \displaystyle y' and substituting out y using the original equation gives LaTeX: y' = \left(\frac{3}{2 x + 6} - \frac{5}{x + 3}\right)\left(\frac{\sqrt{\left(2 x + 6\right)^{3}}}{\left(x + 3\right)^{5}} \right)