plocha=Plot3D[Sin[x+y^2],{x,-3,3},{y,-2,2},Mesh->None,ColorFunction->"Aquamarine",PlotStyle->Opacity[0.7],AxesLabel->{x,y,z},LabelStyle->Medium];
curvex=ParametricPlot3D[{0,t,Sin[(t^2)]},{t,-2,2},PlotStyle->{Blue,Thick}];
curvey=ParametricPlot3D[{t,0,Sin[t]},{t,-3,3},PlotStyle->{Green,Thick}];
curvexy=ParametricPlot3D[{t,t,Sin[t+(t^2)]},{t,-3,3},PlotStyle->{Red,Thick}];
arrowx=Graphics3D[Arrow[{{0,0,0},{1,0,1}}]];
arrowy=Graphics3D[Arrow[{{0,0,0},{0,1,0}}]];
arrowxy=Graphics3D[Arrow[{{0,0,0},{1/(Sqrt[2]),1/(Sqrt[2]),1/(Sqrt[2])}}]];

Show[plocha]

Show[plocha,curvex,curvey]

Show[plocha,curvex,curvey,arrowx,arrowy]

{D[Sin[x+y^2],x],D[Sin[x+y^2],y]}

gradient=VectorPlot[{Cos[x+(y^2)],2y Cos[x+(y^2)]},{x,-3,3},{y,-2,2},VectorPoints->20,VectorScaling->Automatic,VectorColorFunction->None]

GraphicsRow[{plocha,gradient}]