pnew=Sqrt[(Sqrt[p^2+m^2]+de)^2-m^2]

D[pnew,p]

D[pnew,de]

znew=z pnew Sqrt[p^2+m^2]/(Sqrt[pnew^2+m^2]p)

Simplify[D[znew,de],Assumptions->de+Sqrt[m^2+p^2]>m^2]

dznew=D[znew,p]//.{Sqrt[m^2+p^2]->Eold,de+Eold->Enew,m^2+p^2->Eold^2,Enew^2-m^2->Pnew^2}

Simplify[dznew,Assumptions->{Enew>0,Pnew>0,Eold>0}]