I guess you mean "differentiate". Just differentiate each component:
[tex]\vec r(t)=t^2\,\vec\imath-\vec\jmath+2(t+1)\,\vec k[/tex]
[tex]\implies\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=\dfrac{\mathrm d[t^2]}{\mathrm dt}\,\vec\imath+\dfrac{\mathrm d[-1]}{\mathrm dt}\,\vec\jmath+\dfrac{\mathrm d[2(t+1)]}{\mathrm dt}\,\vec k[/tex]
[tex]\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=2t\,\vec\imath+2\,\vec k[/tex]
