Use logarithmic differentiation to find $\frac{dy}{dx}$ for $y=\frac{(3x+1{)}^5{e}^x}{\sqrt[\phantom{2}]{2x^5+1}}$

$\frac{dy}{dx}=\frac{1}{\frac{15}{3x+1}+1-\frac{5x^4}{2x^5+1}}$

$\frac{dy}{dx}=\frac{(3x+1{)}^5{e}^x}{\sqrt[\phantom{2}]{2x^5+1}}*[\frac{15}{3x+1}-\frac{5x^4}{2x^5+1}]$

$\frac{dy}{dx}=[\frac{15}{3x+1}+1-\frac{5x^4}{2x^5+1}]$

$\frac{dy}{dx}=\frac{(3x+1{)}^5{e}^x}{\sqrt[\phantom{2}]{2x^5+1}}*[\frac{15}{3x+1}+1-\frac{5x^4}{2x^5+1}]$