[tex](1+2x)^{x^2}=e^{\ln(1+2x)^{x^2}}=e^{x^2\ln(1+2x)}[/tex]
Differentiate with the chain rule:
[tex]\dfrac{\mathrm d}{\mathrm dx}e^{x^2\ln(1+2x)}=e^{x^2\ln(1+2x)}\dfrac{\mathrm d}{\mathrm dx}[x^2\ln(1+2x)][/tex]
Use the product rule for the rest:
[tex]e^{x^2\ln(1+2x)}\left(2x\ln(1+2x)+\dfrac{2x^2}{1+2x}\right)=2x\left(\ln(1+2x)+\dfrac x{1+2x}\right)(1+2x)^{x^2}[/tex]
