$f(x)g(x)\times k(x)\Rightarrow {f}'(x)={g}'(x)k(x)+g(x){k}'(x)$  $f(x)=({{x}^{2}}+1)\sqrt{x}\Rightarrow {f}'(x)=({{x}^{2}}+1{)}'\sqrt{x}+({{x}^{2}}+1)\times (\sqrt{x}{)}'$ $\Rightarrow {f}'(x)=2x\sqrt{x}+({{x}^{2}}+1)\times \frac{1}{2\sqrt{x}}=2x\sqrt{x}+\frac{{{x}^{2}}+1}{2\sqrt{x}}=\frac{4{{x}^{2}}+{{x}^{2}}+1}{2\sqrt{x}}=\frac{5{{x}^{2}}+1}{2\sqrt{x}}$