Take logarithms of a positive variable power

Take logarithms: $\ln y=x\ln(2x+1)$. The product rule gives $y\prime/y=2x/(2x+1)+\ln(2x+1)$. Substitute back to obtain $\dfrac{dy}{dx}=(2x+1)^x\left[\dfrac{2x}{2x+1}+\ln(2x+1)\right]$.