Simplify a logarithm before differentiating

The square root covers the entire fraction. Write $\ln\sqrt{e^x/(2+3^{x^2})}=\frac12[\ln(e^x)-\ln(2+3^{x^2})]=\frac12[x-\ln(2+3^{x^2})]$. This retains the printed simplification before differentiating. The radicand is strictly positive for every real $x$.

The inner derivative is $d(3^{x^2})/dx=3^{x^2}(\ln3)(2x)$. Thus the derivative is $\frac12[1-3^{x^2}(\ln3)(2x)/(2+3^{x^2})]=\frac12-x(\ln3)3^{x^2}/(2+3^{x^2})$, the printed result, valid for all real $x$.