x[u_,v_]=2Sin[u]Cos[u]Cos[v]/(1+(Sin[u])^2);
y[u_,v_]=2Sin[u]Cos[u]Sin[v]/(1+(Sin[u])^2);
z[u_,v_]=2Cos[u]/(1+(Sin[u])^2);

ParametricPlot3D[{x[u,v],y[u,v],z[u,v]},{u,-Pi/2,Pi/2},{v,0,2Pi}]

dxdu[u_,v_]=D[x[u,v],u]
dydu[u_,v_]=D[y[u,v],u]
dzdu[u_,v_]=D[z[u,v],u]

dxdv[u_,v_]=D[x[u,v],v]
dydv[u_,v_]=D[y[u,v],v]
dzdv[u_,v_]=D[z[u,v],v]

dxdv[0,0]

dydv[0,0]

dzdv[0,0]