θ1=2/6Pi; ϕ1=0;
dθ=3/12Pi; dϕ = 3/12Pi; dr=0.9;
CoordinateTransform["Spherical"->"Cartesian",{r,θ,ϕ}]
$Assumptions=r>0 && Sin[θ]>0;
ss={x->r Cos[ϕ] Sin[θ],y->r Sin[θ] Sin[ϕ],z->r Cos[θ]};
myPoint={r->1,θ->θ1,ϕ->ϕ1};
v0= CoordinateTransform["Spherical"->"Cartesian",{1,θ1,ϕ1}];
e1 = TransformedField["Spherical"->"Cartesian",{1,0,0},{r,θ,ϕ}->{x,y,z}]/.ss/.myPoint
e2 = TransformedField["Spherical"->"Cartesian",{0,1,0},{r,θ,ϕ}->{x,y,z}]/.ss/.myPoint
e3 = TransformedField["Spherical"->"Cartesian",{0,0,1},{r,θ,ϕ}->{x,y,z}]/.ss/.myPoint

Show[
SphericalPlot3D[1,{θ,0,Pi},{ϕ,0,2Pi},PlotStyle->Directive[Opacity[0.5]](*,MeshStyle->Thick*),Mesh->{5,11},ColorFunction->"AtlanticColors",PlotPoints->20],
SphericalPlot3D[1,{θ,θ1,θ1+dθ},{ϕ,ϕ1,ϕ1+dϕ},PlotStyle->Directive[Opacity[0.3]],MeshStyle->Thick,Mesh->{0,0}]
,
(*SphericalPlot3D[1+dr,{θ,θ1,θ1+dθ},{ϕ,ϕ1,ϕ1+dϕ},PlotStyle->Directive[Opacity[0.3]],MeshStyle->Thick,Mesh->{0,0}],
*)
Graphics3D[Arrow[{{0,0,0},v0}]],
Graphics3D[Arrow[{v0,v0+dr e1}]],
Graphics3D[Arrow[{v0,v0+dθ e2}]],
Graphics3D[Arrow[{v0,v0+dϕ e3}]],
Boxed->False,Axes->False,ImageSize->666
]